Obligatory IANAL, etc.

If the template is being used for non-commercial services and does not closely replicate any of the material the characters is based on, then it probably falls under Fair Use - similar to how many rulings have affirmed that fan fiction is broadly legally permitted. Conversely, if the chat service owner is charging for the use, then it would probably be forbidden under the grounds that the service host is financially benefiting from another’s copyright.

Between that, is a murky zone.

• Content creators and owners have at times made legal demands that pornographic or other obscene, shocking, or other fan content which could reflect poorly on the original owners be removed, on the basis it damages their value. If I remember correctly, rulings on this have gone both ways and the issue remains largely unsolved.
• If the bot hoster makes small changes to the identity represented to the bot, it could likewise become iffy. It’d likely depend on the court ruling whether the identity was “substantially changed” enough.
• A new course would be to argue that - given some of the issues regarding how bots have become abusive or encouraging of harmful behavior - a content owner could argue that any chatbot usage represents an intolerable danger to their brand value. I actually expect to see this litigated fairly soon.