It works on Fennec, even in private mode. (Fennec is the F-Droid Firefox build.) Do you have any addons? It works with uBlock Origin on desktop, haven't tried mobile.
I'm on a desktop using FF with uBO, both with fairly locked-down configs, and it doesn't work for me. That is, both websites eventually lead to a broken page because of too many redirects, and the URLs in the URL bars of each tab do not contain the same ID.
Without digging too deep into what your websites are doing, I did notice that they're trying to set / read cookies, which is a) being blocked by uMatrix in general, and b) perhaps being blocked by me having configured FF to block third-party cookies in particular.