keepass seems to be windows-only
The classic 1.x version was tested on Wine and it worked. According the 2.x version works under Mono but I have not tested that. So it is not exclusive to Windows.
What are the specific problems with using lastpass?
Closed-source. You don't know if it works correctly or have no backdoors. It is suboptimal design for password storage. The synchronization is convenient but it is a tradeoff of security.