Is the velocity of a string that is being rotated around a central point the same at any point on the string?

For an object rotating about an axis, every point on the object has the same angular velocity. The tangential velocity of any point is proportional to its distance from the axis of rotation, i.e. $\mathbf{v_{\perp}} = \boldsymbol{\omega} \times \mathbf{r}.$ So it depends what you mean by "the velocity of the string."

For different points of the string to have same velocities, their direction of movement and speeds would to be the same as well. If the string is taut then all points are moving in the same direction at any given time, but it's easy to show that the speeds are not the same: If you have your 2 meter long string, the endpoint will move the total distance of 2·π·(2 m)≈12,6 meters during one full revolution. If you specify the speed of 5 m/s, one revolution would then take 4π/5 s ≈ 2,5 seconds.

Clearly one full revolution takes the same time for all points on the string (if it's considered taut, rigid). But where the endpoint travels more than 12 meters in one revolution, points closer to the rotational axis end up moving a smaller distance as any point closer will trace a circle that is smaller than the one drawn by the endpoint. Thus the points closer in travel a smaller distance in the exact same time and hence their speed is less.

(Linear) Velocities therefore can't be the same.