If you want to get technical, anthropomorphism to the level of art form and dressing up was almost ubiquitous up until recently in the history of humanity. That being said, I don't think it will ever be "mainstream" in the modern world. Something needs to have a ubiquitous and innocuous appeal to those interested, followed by neutrality or indifference from those who aren't part of it. It's the second part that nukes your chances of it being mainstream any time soon. When someone outside of the community hears "furry", they shift their attention to the explicit part. Hell, just type that in the search bar and scroll down until you get something distinctly sexual. I know that is just one part of a larger community, but that is what a wary person will conclude to. It isn't just isolated to the Furry fandom; There was a lasting period during the "Animation Ghetto" that "Anime" and "Hentai" were seen as synonymous, which frightened the people who followed the mainstream "animation is for children" mentality.
it weirds people out, and that is the whole reason me and friends wear fur suits
put a suit on and mope around in public places, get funny looks, good times