Any chance you wrote down (and still have) the ser# of the one you used as a kid?
I've never seen a Mohawk in anything but .243. I have one. Also have Model 600s (not Mohawk 600) in .222, 6mm, .308, & .350Rem Mag.
Any chance that old ammo (.243) has some fired cases??
The 6mm case is 0.078" longer from the base to the base of the neck than the .243 is. .243 cases fired in a 6mm chamber will have the shoulder blown forward that amount, which would be obvious if compared to an unfired case, or one fired in the correct chamber.
The .243 case is also spec'd 0.024" greater diameter at the shoulder than the 6mm case, which might make it a crush fit in a 6mm chamber, depending on individual case and rifle tolerances.
As I see it, the possible options are (in no particular order)..
#1) you mis-remembered the caliber of the gun
#2) its a different gun, same make, & model, but in a different caliber...
(perhaps something happened to it, and your relative replaced it with the closest thing he could find?)
#3) its the same gun, but rebarreled from .243 to 6mm...
(same as #2, but relative shot out the barrel, and could only get a 6mm barrel to replace it with... if either one of these, you need to talk to that relative,
)
I have no idea what actually happened, just naming possibilities....
#4) it is the same gun, a rare factory mismark and you never realized it as a kid...
#5) You are misidentifying a Model 600 as a Mohawk 600
One thing I do know is that you SHOULD NOT shoot .243 ammo in a 6mm chamber, (even if it does work). I don't know from personal experience if it will chamber or not.
If it were life and death, I'd say try it. It's not. 6mm Rem ammo and brass aren't common, but they are out there, still. If you handload, brass can be formed from several other cases, 7x57 Mauser was the parent case for the 6mm (.244 Rem).
I will tell you this from personal experience, if you have a case rupture in a Rem 600, some gas WILL hit you in the face.
If at all possible, some clear pictures of your rifle would be nice. The 600 series rifles went through several variations, with the Mohawk being the last ones catalogued.
Hope this helps. DON'T SHOOT .243 in a 6mm gun!
Take it to a gunsmith and have the chamber checked, that at least will settle if the barrel is properly marked, or not...
Good Luck, let us know more, when you can.