It does seem strange that a host would insist a guest had brought a dog when there was no evidence that this was the case.
@Dora441 in your situation, if you were truly unhappy, I would have asked Airbnb to help me find an alternative listing.
It's a shame that you sent this host a follow up email calling the host unprofessional and left your rather strange review in response to the hosts very decent one. . If you had left well alone and left a neutral review likely this wouldn't have happened.
Sadly if I had you as a potential guest and saw your response, I wouldn't want to host you, not because of what
the host said, but because of what you did.
I would move on.