Teams Only cannot initiate chat to Skype users on Islands mode
When a user is in Teams Only mode, they cannot send a chat to a user who is on Skype. The only way a Skype user can chat with a Teams user is if the Skype user initiates the chat from their end to the Teams user.
Hey, hop into this thread -> https://microsoftteams.uservoice.com/forums/555103-public/suggestions/17765740-skype-chat-both-ways
I think I found a fix for this situation (it includes changing the island mode to Skype for Business with Teams collaboration)