(They could have always searched you by name, if you're worried about your identity then it makes sense to not connect your phone number to FB)
OT, but I had a not-so-nice experience with that. Some years ago I got myself a data-only phone number/SIM card for my 3G dongle.
It turned out that the number was reused, and the previous owner of that number had subscribed to a pay-for message service for upcoming events (concerts, movies, etc). And they kept arriving to my data-only dongle, and I didn't notice for a while. Had to pay for it, though.
I don't remember how I eventually managed to turn that off (the software for accessing/manipulating that part of the dongle was Windows only).