What does non-collusion assumption mean in server-aided protocol based on secret sharing?
Does the "collusion" mean any possibility that t shares in the servers are somehow assembled?


It refers to the possibility that t or more server admins/owners collude to pool their shares to break protocol.

Regarding the follow-up questions: non-collusion is in this context in regards to server operators being malicious. Malicious clients are considered a non issue in the context since clients should not have access to each others shares. Hacking of servers or data leaks is considered out-of-scope since it has nothing to do with the scheme in question.