With a background in mathematical finance and desire to apply for a mathematics PhD in another field, do I need a second master's?
I would say the answer depends on your background in mathematics, and from the information you give, I can't tell how extensive that is.  It isn't so important what the title is on your degree; it matters what you've done and what you know.

To start a US-style PhD in pure math, in which the first couple of years are coursework, you should have the equivalent preparation of a bachelor's degree in mathematics.  It should include, for example, about one year each of proof-based real analysis and abstract algebra.  You should be very comfortable with reading and writing mathematical proofs.

To start a European-style PhD, you are expected to be ready to begin working on research.  You should have completed graduate-level courses in algebra and analysis, and be very familiar with fundamental material in your subfield.  For an area like stochastic analysis, for example, I'd say you'd want to know the majority of the material in a textbook like Karatzas and Shreve, and be able to work through the proofs and solve the exercises.

One question is - what was the level of mathematical rigor in your courses up to stochastic calculus?  If it wasn't high, that's likely to be a problem.