@leocg It's Opera One I believe. Most other sites keep me in, it's just one other I'm having problems with which isn't that big of a deal.
Didn't get a chance to check other browsers, but regardless, I opened it just now to see and the problem appears to have fixed itself...? It keeps me logged in now. Unsure why, I haven't done anything different. Maybe it needed time to register.
Sorry to have wasted your time - but I appreciate your input!!