@Andrea5657 strictly reading between the lines here:
1. You broke things
2. You brought animals who may have made messes
3. You had little ones who may have made messes (and admittedly broke things.)
4. You thought your clean up prior to vacating was fine
5. Your host was agreeable until he/she saw the extent of the damage
6. Your host filed a claim which you probably refused because you thought your clean up was fine.
7. Your host tried to leave an appropriate review
8. You were upset and demanded it get removed. Airbnb complied
9. Your host went off the rails and reached out to someone he/she should not have, maybe to get some kind of compensation, maybe just to vent?
Both of you are in the wrong here. Many times guests feel like the condition they leave a place in is just fine, but when the property owner or cleaning crew reviews, its far from it. $590 may have been steep, or it may have been reasonable for a deep clean after pets and kids and replacing damaged items. I really don't know. Why not try to negotiate to something where the host could get compensation for damaged items? If you felt $590 was too high, what did you think was reasonable to replace the mirror and assist with turnover?
As for someone finding out about you on social media, did you book with a shared account? Any chance your husband's name was anywhere on this reservation? Not excusing your host's behavior but I do wonder if there was something obvious they saw in the booking that they reached out to, perhaps in desperation for assistance with the damage/cleaning fees.
I would advise that in the future you stay in hotels where there is a specific protocol if something is damaged or missing. Far better to avoid this sort of drama. Best of luck.