Ok I'm stumped. 

Where did you find it. I have same motor - not seen that.

It could be fuel starvation. Sounds like you know what you are doing.

It looks like a small inductor which is used as an electrical filter on something like an old tv or radio. Is the copper colour a fine copper wire wound around a ceramic spool? I can't say I've ever seen one on an outboard. If it is that then my guess is its from something else that's found it's way in there.

I have had near everything off my Merc (s) 90 over the years. I don't believe it belongs in there.

Can you check your main charging fuse, the photo may not look like a normal fuse but the engine not running above idle can point to a fault in the charging system. There’s a main fuse in the regulator side and if this blows or the regulator is faulty the charging system will produce to much voltage for the coils and won’t run above idle.

On later model motors this fuse was replaced with a fusable link. Older motors had a 50 amp fuse which did look at bit like this.

Just to check you haven’t replaced the battery with the wrong type of battery like a gel cell or a house battery, this can damage the regulator as these batteries need to be changed at a lower rate than a standard start battery.

Hiya, I have had very similar symptoms with a 40hp ELPTO (two stroke) mercury. As soon as I put it in gear it would stall, or move forward briefly and stall. I managed to drain the carb while on the water, which was enough to get me back to the ramp. I put it down to water in the fuel or stale fuel. I got the fuel from the campground tank. Have you tried draining the carb, and connecting up a fresh tank of gas? The gizmo you found could be a red hearing, have no idea how it got there or what it is. Smudge has really good knowledge of electrical systems etc, and has stripped a similar motor so I would count that out. Worth a go anyway....

Thanks guys,