Berlin is a state because (esp West-)Berlin kind of had to be one in the restructuring after WW2.
Bremen and Hamburg both were Free Cities in the Holy Roman Empire. More importantly, they were among the very few Free Cities that retained this status until and past the end of the HRE (1806). Being in the Hanseatic League helped establish their importance, but the League wasn't actually a major concern at that time anymore, and hadn't been for over a century, but they certainly played to that history (and e.g. added "Hansestadt" to their names, long after)
Bremen and Hamburg both were Free Cities in the Holy Roman Empire. More importantly, they were among the very few Free Cities that retained this status until and past the end of the HRE (1806). Being in the Hanseatic League helped establish their importance, but the League wasn't actually a major concern at that time anymore, and hadn't been for over a century, but they certainly played to that history (and e.g. added "Hansestadt" to their names, long after)