If a monster has multiple legendary actions to move up to their speed, can they use them to move their speed every single turn they use the action?

This particular monster, let's call him Strahd, has 3 legendary actions. One of them says:

Move. Strahd moves up to his speed without provoking opportunity attacks.

Let's say he has 50ft of movement. Does it mean that he could move up to 150ft in one round with just legendary actions? Or would he be limited to move up to his speed, so that he could only move up to 50ft in total in one round?

The rules on legendary actions simply state:

A legendary creature can take a certain number of special actions — called legendary actions — outside its turn. Only one legendary action option can be used at a time and only at the end of another creature's turn. A creature regains its spent legendary actions at the start of its turn. It can forgo using them, and it can't use them while incapacitated or otherwise unable to take actions. If surprised, it can't use them until after its first turn in the combat.

Nothing prevents the same legendary action from being used more than once per round (though only one option can be used at the end of a given turn).

If there are at least 3 other creatures in the combat, this means (up to) three 50-foot moves in your example. Strahd should always bring friends.

The rules for Your Turn say (emphases in the original):

On your turn, you can move a distance up to your speed and take one action.

Later, the rules for Movement and Position say:

On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn, following the rules here.

It is true that with a naïve reading of "you can move a distance up to your speed" one could infer that your speed, recorded on your character sheet, sets a maximum distance that you can travel "on your turn". That is, this is a hard limit per turn, and you can't move farther under your own power.

However, that is not the intended meaning. Rather, what these rules are trying to say is that you can move up to your speed with your movement. On the same turn, you can additionally move up to your speed with your action, if you take the Dash action:

When you take the Dash action, you gain extra movement for the current turn. The increase equals your speed, after applying any modifiers. With a speed of 30 feet, for example, you can move up to 60 feet on your turn if you dash.

Any increase or decrease to your speed changes this additional movement by the same amount. If your speed of 30 feet is reduced to 15 feet, for instance, you can move up to 30 feet this turn if you dash.

This makes it clear that the per-turn limit to the distance traveled is not your speed, but rather your speed for each ability that allows you to move - in this case your movement and your action. You could even move three times your speed in a turn, by using your movement, your Action (Dash), and your bonus action - if you have a feature that grants one (eg. step of the wind for monks, cunning action for rogues, or the expeditious retreat spell for at least five different classes of casters).

Seen through this lens, legendary actions are just another way to gain extra movement on your turn, and each legendary action spent allows you to move up to your speed, but there is no hard limit on your total distance moved per turn (and see many questions on this site trying to maximize distance moved with complicated builds).

The legendary actions part of a monster's stat block describes how they can be used. The one for the actual Strahd says:

The Strahd von Zarovich can take 3 legendary actions, choosing from the options below. Only one legendary option can be used at a time and only at the end of another creature's turn. The Strahd von Zarovich regains spent legendary actions at the start of its turn.

Nothing in here prevents using the same legendary action multiple times except the hard limit of three actions. However, this is not always the case. Tiamat, for example, gets five legendary actions. Using a breath weapon from one of her heads costs 2 legendary actions - so she could breathe with her red dragon head (her best attack) twice between turns, except that her stat block also says:

Once Tiamat chooses a legendary action option for one of her heads, she can't choose another one associated with that head until the start of her next turn.

This text limits her use of the red dragon head to once between her turns, despite her having legendary actions available. If Strahd was meant to have such a restriction on repeatedly using his Move action, his stat block would say so.

You say "This particular monster, let's call him Strahd," but also give his Speed as 50 feet. Are you talking about the Strahd? Because he has a speed of 30 feet (walking as a vampire, or flying as a bat) or 40 feet (walking as a wolf).

There is no hard limit (other than action economy) on the number of times per turn or round a PC or monster/NPC can move their speed. Using all your actions + legendary actions on movement (dash, etc.) can indeed produce big numbers in physical feet/second speeds. Combined with ways to boost your speed stat, things can get pretty silly, to the point that a DM might want to step in and apply a reality check, although for creatures like vampires with supernatural powers, really fast movement isn't immersion-breaking (especially when not sustained for many rounds, which should be tiring.) Unlike a supersonic tabaxi rogue which strains realism for some players... (Pushing 5e mechanics outside the regimes they're designed for can often give weird results that don't match physics.)

The connection between your speed stat and how far you can actually go in a combat round is interesting to try to explain narratively; in case that's what made you wonder about this, I'll try to answer since it's something I've thought about before.

In my understanding of the connection between mechanics and narrative, it's important to keep in mind the assumption that all creatures participating in combat are aware of everything happening around them. (Including locations of invisible creatures that haven't taken the Hide action, based on sound). And you're defending yourself.

When you move, you're not just turning and running as fast as you can, you're moving as fast as you can while maintaining situational awareness on top of whatever else you're doing on your turn. (Unless you disengage, though, you are giving opponents extra openings to attack you, but not with advantage so it's not quite like you're turning your back on a sword-fight and walking away.)

Narratively, the Dash action is spending more of your focus and time on just moving (and maintaining awareness while doing so), instead of also trying to fire arrows or cast spells while you move. Narratively, in some turns where you move your speed and then take an action other than Dash, you weren't walking for the full 6 seconds of the round, you moved quickly then engaged in some swordplay, or lifted a heavy barrier, or searched a desk... Or in other turns, you weren't giving much of your attention to your footsteps as you cast a spell or aimed arrows but were moving fairly continuously for most of the 6 seconds. And all of this while keeping your head on a swivel to defend yourself (getting your dex bonus to your AC) from melee and ranged attacks from all directions, and ready to make a dex save against any spell.

Rogues and monks being able to Dash as a bonus action is a mechanical representation of their fast reflexes and/or mental quickness in combat: they can maintain situational awareness even while moving faster and/or doing more stuff than other characters. And if they put all their attention toward movement (Dash as their action and bonus-action), they can move farther than others while still defending themselves and maintaining awareness.

Legendary actions that allow movement can be thought of similarly, if they're not teleports.

5e has separate rules for chase scenarios where the basic assumption is that you're putting most of your focus on movement and aren't having to maintain 360 situational awareness and defend yourself. Rogues aren't (in my understanding) supposed to be 50% faster than other classes (3x their speed instead of 2x) in general sprinting, like in an athletics competition like a 400m race (1 lap around a track, 1312 ft). (Athletics is the name of the track & field event category at the Olympics etc.; not a coincidence that D&D has a skill of that name.) Possibly over shorter distances like 100m (328 ft) where fatigue isn't a factor, it makes narrative sense for a rogue to have some advantage over a wizard (without using magic) even if both have the same Con score and same Str(Athletics) modifier. But you could really go either way on this.

Anyway, you don't have to use combat movement rules outside of combat, for example you could roll athletics checks for a sprint race, or Con(Athletics) over longer races like the 1500m or marathon. Combat movement rules are a reasonable default most of the time, though, since consistent gameplay is usually more important than realism of characters sprinting to join a combat that starts some distance away from them.

(The DM should create a scenario balanced around the movement rules they're actually going to use, i.e. standard combat movement, so the PCs or extra enemies can arrive at a suitably dramatic moment if that's the desired outcome. Then the narrative in people's heads is still of PCs fully sprinting to help, even though their movement speeds were only 60 ft/round (10ft/sec = 11 km/hour) for most of them, from movement+dash. According to https://marathonhandbook.com/average-human-sprint-speed/, "most non-elite adult runners can sprint 100m" in 12-20 seconds, for a speed of 18 to 30 km/h. (It's unclear if by "runners", they mean people who regularly run for exercise, but maybe.) That's 98.4 to 164 ft/round. Of course, that's fully unencumbered, wearing modern running shoes. Realism and simplicity of game mechanics are often at odds.)