Yes, Marriott has two group billing processes: Master Billing is used for group reservations where all the billings (mostly just room & tax) is rolled up into one bill and presented to the company. The other: Direct Billing is used for individual reservations that are billed separately to the company. Master Billing is not supposed to earn credit for the stay, whereas Direct Billing usually does.

Since you can see your reservation online, it is my guess that you are on a direct billing arrangement. Contacting the hotel to add your MR number should cause you to earn credit for the stay and that's legit. If someone at the company has added their MR number to the rez, they should NOT be getting credit. The only exception here is if they are a meeting planner, in which case you both should get credit.

Years ago there was a big push to get travel credit issued to the company and not the person travelling. And there were threats to make such credit taxable in the US. Fortunately that go shot down and almost all programs now state they own the credit and can do pretty much whatever they want with it. And right now, we are the ones allowed to use that credit, be it personal or business use, our choice.

Yes, a company can have a policy stating that credit earned on business trips must be applied to business travel (probably free rooms and not upgrades). In which case I would probably not belong to any of the programs. But, I'm not too sure how they would know unless you self-report.