Should the support structure for a kitchen island be built on top of tiles?
No. 
The tile beneath the pony wall should be removed.

One of the things that a thinset and tile underlayment do is decouple the tile from the subfloor beneath it. By contrast, you explicitly want the pony wall supporting the slab to be coupled to the subfloor.

Then the pony wall should be secured both to the subfloor and the cabinets against it.

The sub-counter (I recommend using MDF) should then be a single piece attached directly to the cabinets and the pony wall.