Yes you can, I have done it.
The core O/S is the same.
The SysID module is the same.
The customer database format is the same
The hardware platform of the MX controller is more robust, supports more users, faster processor, etc.
I believe you can also move the hard drive over if you're so inclined, tho I wouldn't.
Just make sure you have a good datasave (I always make two just to be safe, in two different locations) then power-down the 100-user controller.
Pull the cover off (per the procedure in the Tech Hbk (that one is a little tricky) then pull your SysID module (1 screw) and move it over.
Any MMC boards in the 100-user will move as well, i.e., Dual FIM.
Power the MX up and then perform a MANUAL software install (again, per the tech handbook).
Once you have the software installed then perform a data restore from your saved database.
Upon completion of the data restore the box should perform a spontaneous reboot and then come up.
All in all you're going to be down a couple hours start to finish.
Just me, there's enough hoop jumping going on here without complicating things with AMC.
Therefore I probably wouldn't attempt putting 6.1 on the new box until after you get it up and running with the customer's database and let it run a week or two.
Something else, don't try to be a hero by trying to jump the customer's database from a 3-dot load all the way to 5-dot or 6 in combination with this hardware upgrade.
Elephant is best served one bite at a time; if you want to be at 6.0 on the new box then I'd upgrade the customer to 6.0 now (on the 100-user controller) first before moving him across so you can do your datasave at the same O/S rev level you'll be introducing the database to on the MX before.
If you're jumping multiple loads in this upgrade be sure to review the Release Notes on all the interim loads.
I know Mitel sez this hardware switcheroo in combination with a multi-stream upgrade/data convert all in the same breath will work, but I sure wouldn't try it.
I would upgrade the software first, (on the 100-u) let it stabilize a few days, then get a datasave and then do the hardware exchange.