If the face frame of the vanity will sit tight to the sidewalls for a built-in look, there's no real need to tile the side walls. With that installation, I'll rock the walls, set the vanity, set the top, then tile the upper walls.

The tile can over a micro-gap between the countertop and the drywall. If the countertop is slightly wide for the wall-to-wall dimension and the countertop can not be cut down, I'll score and notch the drywall, like making a groove in the drywall, then slide the countertop into the grooves so it fits nice and snug.

Then tile.

There's nothing wrong with tiling the walls then sliding the vanity into the space. That is sometimes done to "future proof" a room regarding vanity replacement down the road. But you might not get a clean detail where the countertop abuts the tiled walls.

More often when a vanity is set into an alcove with three fully tiled walls, the vanity is not as wide as the alcove. There's a roughly 4" gap on each side between the vanity and each side wall.