I currently have a Humminbird 535.  If I want to replace this unit with, say a Lowrance, can I simply unplug the old unit and plug the Lowrance into the existing transducer?  Or does each brand/model have it's own transducer?

Proprietary plugs.

Each has there own not even all same brand models work with the same transducers) .... A PITA if you ask me.  >

And do not cut the end and try to splice on another connector, the length of the cable adds to the impedance of the unit.

You cannot swap tranducers for one brand to another.

There are universal transducers. Vexilar has one for aluminum boats and Minn Kota and Motor Guide have motors with built in transducers. In each case an adaptor is needed to match the unit.

Lowrance and Humminbird offer adapter cables to use older transducers with some of their newer units.