The first time I described my social circle to someone as ‘mainly pagans, perverts and morris dancers‘ was about 20 years ago and, well, not much has changed.
Just recently I went to have a bit of fun that involved expressing the non-smut-focussed side of myself, spending a weekend in the Midlands at the Unconvention of Fools and Beasts. For those of you with some vague idea of what Morris dancing involves (which should include any of you who saw that magnificent performance at the Brit Awards which featured Boss Morris), beasts and fools are often a big part of it.
Dressing up and roleplaying as animals is an absolutely massive part of human culture, pretty much globally, for all sorts of reasons. Christofascists’ fixation on ‘litter trays in schools’ is just another manifestation of their spiteful ignorance, of course and need not be considered here.
I had an absolutely cracking time belting around the centre of Birmingham in my own costume, seeing what everyone else was doing and, yeah, posing for endless photos with passers-by who had stopped to watch. It was also interesting to chat with other beasts. I didn’t detect any particular affinity with furries, but most ascribe some sort of separate personality to the beast itself, irrespective of who might be wearing the outfit at any given time. It might be worth a mention, in passing, of the whole sports-mascot phenomenon, which is a similar world-of-its-own, with people who portray a club’s mascot being quite invested in their role, though I can confidently state that none of the Morris lot take it this far.
Cosplay, I think, is a slightly different category in that a lot of cosplayers have new outfits and personae for every event and/or like to dress up as a known and identifiable fictional character rather than developing a specific creation with a life and backstory of its own. 
However, as you will be hearing rather more on a topic that is at least very closely aligned with this soon, why not have a think about what kind of beast you might be, given the option?