Conservation laws are strongly dependent on what reference frame you are working in. A reference frame consists of everything one single observer would measure. Any physical situation can always be described by one observer who considers himself stationary. When writing down conservation of linear momentum, you should only consider the velocities that particular observer would see.
In this problem, the velocity $v$ of the plate is given with respect to the ground. This means there is an observer on the ground who measured its velocity. However, the velocity $v$ of the man is given relative to the platform, so there is a second observer on the platform who measures the velocity of the man.
When applying conservation laws, you need to convert all available values to one reference frame (as would be measured by one of the available observers). You cannot use the velocities $u$ and $v$ as given because they were measured in different reference frames. If you want to use the velocity $u$ of the man relative to the plate, you are working in a reference frame where the plate is stationary (because $u$ was measured by an observer moving with the plate who considered himself stationary). In this frame, the ice would be moving instead. So you would have to balance out the momentum from the ice and the man, instead of the momentum from the plate and the man. However, the mass of the ice is not given, so this is not possible.
The sensible solution then, is to convert the velocity $u$ as measured by the moving observer to a velocity $u'$ as it would be measured by the observer on the ice.
TL; DR: Conservation laws are only valid when all involved quantities are measured in the same reference frame (by the same observer). This is not the case in the present problem
I hope this was clear. I know the concept of reference frames and the switching between them was not trivial to grasp when I first encountered it.