a second trim switch, either in the throttle handle, or a trailer button

They may not go to anything at all if the guy previous guy was too lazy to pull old wiring out when re-doing stuff. They look like the standard plugs that go into trim/tilt pumps...the purple wires would connect to the limit switches (the hockey pucks on the outside edges of the gimbal assembly) so the pumps know when to stop.
If everything works on the boat then either leave them alone or yank them out and clean things up....you can test them to make sure they not hooked up to the switches. Just place the ground lead of a volt meter on any good ground, then the positive lead into either the green or the blue wires and start hitting trim/tilt switches up/down to see if there's any power going to them...

Yea, there were NO old wires pulled anywhere. Wiring is a mess I think you have just hit it. The "trailer" switches for the drives were replaced with aftermarket. There are no "trailer" switches, only tabs and regular trim plus throttle trim. So what you said about the trim limit switches makes sense, as there is no "limit". drives trim all the way with button on throttle or just the seperate trim buttons.