In the US, and in most other places, you can't make "coins" and sell them without being accused of fraud (or counterfeiting), but there's no such protection on the word "mint". You can start up a company with "mint" in the name, and no-one really cares or objects - unless the name is already registered. Many such private mints around the world strike medals, bullion rounds, tokens, badges etc, and some are even big enough and reputable enough to obtain contracts from real governments to mint real coins (eg. Pobjoy Mint in Britain and
Perth Mint in Australia).
At the other extreme, some "mints" don't even actually make anything, they just repackage and on-sell coins struck by others. One such company here in Australia is the "Macquarie Mint"; their business model is to purchase bulk lots of ordinary coins produced by other mints, produce or find a special collectable "go-with" like a print or a badge, re-package the lot in Macquarie Mint wrappers and sell the coin and "go-with" for a considerably higher price than the company paid to buy it.
As for mintmarks, if you own the mint, you control the design processes and set company policies. You can put whatever design you (or your customers) wish on the things you make, and you can call part of that design a "mintmark" if you like.
Owning a mint isn't as profitable a business as it used to be, because people are finding it cheaper to outsource their medal / badge / token / bullion round production overseas. Here in Brisbane, Australia, there used to be half a dozen "private mints". There's only one left now that still does some of it's actual striking here, all the rest now outsource to China.
Don't say "infinitely" when you mean "very"; otherwise, you'll have no word left when you want to talk about something really infinite. - C. S. Lewis