Hello @Alexey and thank you for your replay. No need for check by you but thank you for the offer.
I found out that the problem is in Messenger settings ( /studio/settings.php?page=bx_messenger) where we can choose emoji style.
And althoug there is no more "emojione" we can still choose it in the emojie settings. And there are more of them - I mean we can choose more emojie styles with no coresponding set in the path mentioned in my first post.
Our UNA ver. is 14.0.0-A2 and Messenger is version 13.0.1 - I just checked and I see there is an update to 13.0.2. Will update now and I will get back with findings after update.