Bathroom Vanity against a pre-existing wall of tile

You should find studs behind the tile and whatever backer board was used. You may find that the current vanity is attached to these same studs.

This vanity should be screwed to the wall. Open the doors and look at the rail near the top of the vanity on the back. Don't just yank the vanity off the wall. You'll attach the new vanity the same way. You drill through the tile and insert a plastic molly on each end. You insert the screws through the back rail, into the mollies and screw them snug to the wall. You might be able to use the old mollies if you can drill holes in the new vanity that match the molly configuration. To do that place one nail in each hole, point out. Position the vanity where you want it and slide it back so the nails leave a mark on the rail. Pull the vanity out and drill an over size hole to give you some leeway. Insert a screw with a washer through the hole and push the vanity back enough so you can get the screws into the mollies. Start the screws and then finish pushing the vanity to the wall. Tighten the screws so they're snug.
Ron