Does velocity add like vectors?

In this problem, you are not concerned with the vector sum of the two tensions, only with the relative velocities. For that you can work with just one side.

This kind of diagram is often used to study forces. Forces do add like you are thinking.

But this is velocity. Consider the case where $Theta = 0$, and the pulleys are right next to each other. If P and Q drop as distance, you would expect M to rise the same distance because the string stays the same length.

Forces can't like vectors however the velocity does not.

The velocity of the block will be the same as the velocity of the piece of rope attached to it. The velocity of the vertically hanging strand is equal to the rate at which the whole rope overhanging a pulley is moving up.

If the $ 2v cos theta$ was the velocity of the block, then it would mean that the block is moving faster up than the rope is going up!

Comparison with boat problems, in the boat problems it's like the river acts like an escalator (Know the one in airports?) so that either boosts the speed of the boat or reduces it. Here it is different because the object is constrained to move at the same velocity as the rope for a physical situation. THe closest analogy I can think of is a paper boat moving at the same speed as a river flows.

You see the addition of velocities is done when you are changing reference frames, but in this case the velocity of the mass is the upward velocity of the point to which it is attached. Hope this explains your doubt