How to fix a slate floor tile in my kitchen
Looks like slate

It isn't.  Slate, being a natural product, isn't a different color on the back than it is on the top.  What you have there is a ceramic tile that is glazed and textured to look like slate.  That's actually a good thing in this case because slate is fragile and often requires more care in order to get it set level.

What's the worst that can happen?

Well...I don't know how it works in the UK, but if you try to repair it and do a poor job, can the letting company come after you for your deposit?  It might be more hassle later on than it is just to have the landlord get it fixed.

What is this adhesive called?

What you're looking at is called "thin-set mortar", and it is a cementitious adhesive that usually has to be mixed the way you'd mix concrete.  The "stripes" are troweled into the thin-set using a notched trowel and the tile is set in place.  It's not "adhesive" in the sense that it comes in a caulk tube and you can just squeeze it onto the back of the tile and glue it down.