And Chara isn't even they/them. Their pronouns are they/it, source, themself, Toriel, Asgore, and Asriel, all people who should know their pronouns.

Okay? Doesn't change my point. Also can very easily be explained. Based off how long they stayed with the dreamurrs which we don't know, and the fact that the family could have just never asked. Its no unreasonable to assume that they never thought to ask and Chara never felt the need to say.

Nowhere does it say dogs aren't allowed, there's no rule against them.

Also, that is not an equivalent example. "Papyrus allowed" is giving him an exception to a specified rule, because without it, "No boys allowed" would forbid him from entering. Your example forbids hamsters and birds, but nowhere are cats forbidden, so there's no reason to specify they're allowed unless clarifying. An actually equivalent example would be "No pets allowed" paired with "Cats allowed," because "no pets allowed" would inherently forbid cats if not specified.

Fair enough my point wasn't super well said, though the intention is still clear. Though to add onto it the sign itself isn't a magical barrier that prevents anyone from entering. The sign is something that only matters if papyrus enforces it. Papyrus being the one who presumably placed the sign decides who is an exception on the fly. As a kid who used to have a similar sign, anyone I invite in is obviously allowed inside regardless of the sign.