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).