OK now we're getting places.
Looks like that existing tile is installed over tile backer board which is a typical installation and you would do the same. Not a problem I can see.
Here's the problem I do see:
previous owner left behind some spare tiles with bar-code stickers on them, just a quick call to the local Lowes solved that problem
Those bar codes mean nothing. I doubt the tile is going to be a perfect match but you may have gotten lucky if you haven't seen a shade/color variance thus far.
You need tile backer. It gets installed with unmodified thinset mortar and fasteners according to the specific backer board manufacturers recommendations. The tile can then be installed using modified thinset mortar. Use the proper trowel for both. Everything should match up pretty close.
If tile continues were the metal edge is now you will have to sacrifice those few tiles to remove the metal edge, it is trapped under the tiles. Once you have removed those tiles you can add your adjacent backer board then pick up on the same layout and go from there.
Remove all the grout surrounding those sacrificial tiles before you try to remove them. The tile base baords can stay in place but remove the grout at the floor/wall juncture also before trying to remove the tiles.