How feasible would it be to transition from a Master in Finance to a phd in Pure Maths? I know this sounds weird. I'm getting a Master in Business (MSc, not MBA) with emphasis in Finance (in my country you can't officially get a master in finance, logistics or marketing, they are all called Master of Business but still different based on what you want to research for the thesis).

Thing is, I really became interested in research and academia recently and have been thinking about changing my plans completely (I was working in business management before starting full-time grad school and wanted to go back to the industry, but not anymore, I found out that I like academic research a lot).

I have two college majors, one in Business and one in IT. Do you think it is possible to start a DSc program in Europe in pure maths with this background? Would I have to go back to 0 and get another, more closely related, college degree or maybe masters? I just don't want to get my hopes high and invest time and effort pursuing something unrealistic. I'll finish my masters when I'm 26 next year, if I had to go back to college for a degree in maths I'd be 30 by the end of it. Is that too old for a career as researcher?
Bottom line is that it will be very hard since you lack a bachelors in math (or even the math training of a physicist).

My advice would be instead to go work for a while.  It seems like we get a lot of degree change requests which are prompted from not wanting to leave the nest.

If you don't follow the go work advice, I would at least look for something a little closer to your training.  Something in quantitative finance or econ or at least applied math or statistics.  Not pure math.