I use my real name for Instagram and FB (and on my non-art shop site for the past 18 years). I have separate accounts on FB for art, personal, and non-art shop, but the one that gets all the traction is my personal page, so I post my art stuff on there too and try to keep my rants to a minimum.
On Instagram I have only my art stuff. I just really don't want to post about my personal life on there, even though I know people on there from FB. I actually don't post much about my personal life anywhere online anymore.
I've been online since the 90s and mostly have used my real name. Only twice has anyone threatened to come to my place and do me harm. Both times I said go ahead; I will meet you with the police, who will arrest you for stalking and making terroristic threats (okay, one time I said I would meet him with a baseball bat). Which was the end of that baloney. However, it's a good idea if you own domains, have your site registration under a PO box or pay to keep it private. I don't do either, but if I were a woman I would. Just keep in mind that a lot of people talk a lot of trash online, but I have not heard of one single instance where they carried it out. If it's an ex or something, that's another story.
Using my real name is important for me in terms of my art and also in terms of a book I wrote that is on Amazon.