Master's degree in a different field while in a PhD program?
The question you should really ask yourself is: What is the purpose of having an extra degree? Is it to enlighten yourself and learn something new or related? Is it to improve your CV and future job prospects? 

IMO, looking at your situation, it seem a bit of a waste of time and a lot of extra effort to also pursue your math master. You can study math  on your own accord at any time with the best universities and lectures using the edX free and open university. 


Q: Would I have to apply and be accepted to the Master's program at my university? 


A: Probably not.


Q: How likely do you think this is?


A: Most universities allow for completing multiple degrees concurrently as long as your academic records are in good standing and you pay the additional costs.