I don't think those are necessarily related.

Furries are just people who like animal mascot type characters and made a whole subculture around that.

The dog mask people are generally not furries. They're called pups, and you're actually right that pup play is a BDSM thing. The whole acting like a dog thing is more for dehumanization than anything else.

Furries are people who enjoy anthropomorphic animals for... well, a variety of reasons. Fursuits are extremely uncommon because they're expensive as fuck, difficult to clean, easily damaged, etc. Most furries just come up with fursonas (generally online animal personas) and make art.

I know more about this than I otherwise would because I have friends who are pups and others who are furries.

basically, its a subculture of people who like anthropomorphic (i.e. humanized) animal characters, like zootopia, for one fairly recent mainstream example. some furries do dress up in costumes, but the leather dog masks are a somewhat unrelated bdsm thing, though there's probably some significant overlap in the groups.