There’s no prohibition against owning a mach-capable airplane. And there are lots of them in civilian hands. There is a private flight team that flies F-104’s, which are one of the fastest and highest-flying jets in existance. The F-104 can do about Mach 2.5, if I recall correctly.
Jacqueline Cochran set all kinds of civilian records flying her own F-104 in the 1960’s, at a time when it was still a front-line fighter. I think she had hers built from pieces of scrapped aircraft.
There are plenty of rules limiting Mach flight. Basically, you have to be over the ocean to do it, as I recall.
If you need a type rating for a military jet, it’s a new regulation. I remember lots of talk about that a couple of years ago, because of the high accident rate. But I haven’t followed the issue since then.
Anyway, even if you do need a type rating, all you have to do is find someone who owns a two-seat version of that particular aircraft and get typed in it. So you don’t have to be military. And most popular military jets have two-seat variants available for private training, including the F-104.
That said, a military jet is a maintenance nightmare. They cost thousands of dollars per hour to fly, so they are definitely for the rich. There are a few cheaper jets around, mainly jets which were used for training like the Fuega, the Cessna T-37, and the Canadair Tutor (the airplane the Snowbirds fly). These trainer jets are much, much easier to fly and maintain, and might ‘only’ cost a few hundred dollars per hour to fly and maintain.
And to be able to fly one safely, you’d better have a few thousand hours of high-performance aircraft time, including a lot of jet time. And then you should be prepared to fork over tens of thousands of bucks to the guy who owns the two-seat version so you can get trained in it (even if you don’t need a type rating, it would still be suicide for a private pilot to strap into something like an F-86 or F-104 without a LOT of training).
The initial purchase price is cheap - I’ve seen lots of jets sold for $100K or less. But so what? To fly the thing safely, you’re going to need a full-time crew chief familiar with jets like that, lots of spare parts, a hangar, and the willingness to spend money when you have to (i.e. you can’t get away with leaving repairs until the next annual, etc. If an engine shows signs of dangerous wear or damage, pull out your checkbook and write a cheque for probably more money than what you paid for the plane in the first place…).