When operating the motor with muffs motor pees like it should. 

Should water be coming out of the hole the shift rod goes in the gear case?      red circle in photo.

Man did I have some water circulation problems with my merc 50, although mine is a 2008 4 stroke EFI and looks a little different. I had the same problem through 2-3 water pumps. There’s an intake mechanism that looks like a small pvc pipe with a fluted end. every time a couple different mechanics put my lower back on it got slightly sideways. The motor would still pee Most of the time in neutral and all the time with some throttle but would alarm at very low throttle in gear (trolling plugs). Once a friend of a friend got it back on straight, it shoots water out like crazy at all speeds. I replaced a good water pump for nothing a few times but oh well they are cheap haha. 
 

the first thing to check on all water pumps is poke some weedeater string in the pee hole to knock the dirt dobber nest out 

 

also for all your outboard motor needs, @Way2slow is the authority 

Not true, I don't know much about Mercury motors and don't know if what his is doing is normal or not, so I haven't responded to it. 

Since I have just worked on a 2006 90hp merc, I do know there is a seal that slides onto drive shaft and over the top of the water pump that's probably suppose to keep it from doing that, but if it does or not, don't know since I've not had one on hose and seen where all the different places the water comes out.   

If the water pump hasn't been replaced in the past few years, it would not hurt to replace it and that seal on the drive shaft.  On this 2006 motor, that seal had dried and split, so it would not have sealed if it was running.