The PlayStation 2 does not have a dedicated built-in mouse port. Its front-facing ports are exclusively for:
A few games (like Unreal Tournament or Command & Conquer: Red Alert Retaliation) supported USB mice, but only via the console's USB ports on the back (on fat models) or via specific adapters plugged into controller ports. This was never a standard system feature.
Bottom line: You cannot simply plug a standard mouse into a controller port and expect it to work system-wide. You need a game that explicitly supports it and usually a compatible USB mouse or a specific controller-port adapter.