I have heard of independent journalists, but are independent mathematics researchers a thing that exists?
I am asking because I am intensely interested and invested in a certain branch of maths and feel like I am capable of proving something new, however I am 23 with no formal education beyond highschool and with no interest in going to university.
I am not doing it for the money, I have a stable source of passive income, so I have a lot of free time.
If money is not an issue, then you can absolutely be an independent researcher in mathematics. One of the main reasons of being affiliated with an organization like a university is that you need someone to pay you. In many areas of mathematics you don't need any special facilities or equipment to do your research, so if you don't need an outside source of money then you can by all means do research without affiliation.
However, without some degree of university education it's difficult to know whether your work is correct. In fact it's difficult even with a PhD. It's not technically required, and you could theoretically be a person where everything fits together just right so that you are able to know what you're doing without much guidance (think Ramanujan), but it's highly unlikely. So I would be very cautious if you decide not to get a degree and enlist extensive help from other mathematicians.