It's a Logitech M720- a wonderful mouse that I love a lot. It works with my phone, and personal PC (the one that has Debian on it is a family old one), and for some reason no mater what I do I can't connect my mouse to it.
Out of curiosity, since you mention that the Debian box is an old family PC, how old is the Debian install on it? The M720's spec page shows that bluetooth and the Unifying receiver are supported as of kernel 2.6, which released in 2003. Looks like Debian 4.0 "etch" from 2007 was the first one to use that kernel. I feel like even that's still too recent for it to not be able to be used though, at least over USB