Removing tile without breaking the surrounding tile can be done if approached with care.
In this video, I show how to carefully remove the grout with an oscillating tool and how to cut the grout caulk so that you can safely remove the tile and backing without harming the floor, wall, and abutting tile.
If you need to demo out tile to get at a plumbing leak or other problem this method will help you do it in a way that allows you to reinstall tile that will closely match the existing tile.