How can I become a mathematician with this background? My interest is in mathematical physics.
The only way is to complete a PhD in math (or a field close enough) if you really want to be a mathematician. (There are extremely rare exceptions such as S.Ramanujan.)
From your post, maybe you are not sure if your academic background is enough to pursue a PhD in math. Without more information, I don't think anyone in this forum can really give you a definitive answer on whether you have what it takes to become a mathematician. Your math instructors might have a better answer for you. The truth is, even if you have a "strong" undergraduate math degree, it is still impossible for anyone to give you the probability of success in getting a PhD in math because the road to a PhD depends on way too many variables.
However, not to be too pessimistic, I encourage you to apply to a couple of math doctoral programs in the USA and see how things go. You should google "usnews mathematics ranking". With only the information you have given, maybe you do not want to apply to the top 10. Maybe pick one that is ranked 11-15, one from 16-20, etc. This is only a suggestion. You definitely want to talk to your instructors and get input from them. You will need three very strong letters of recommendation from them anyway.
Kudos to you for studying some math topics on your own. Keep at it. Talk to your math instructors for guidance.