No, people don't have two religions, that's a really odd thing to think!
Some people follow aspects of multiple faiths, but then they're either spiritual or loosely 'christian' but its up to the particular faith whether they would recognise them.
Most schools would require baptism into their faith, and attendance at their services - they would acknowledge the other religion as christian, so you wouldn't need to be baptised again, but you would need to convert to be considered a full catholic and to be officially allowed to receive communion as has been pointed out.
I know friends who are atheist (based in Ireland) who had to deal with an enrolment criteria at a catholic school that went: first, catholics - then CofE (or in this case CofI), then, other christian denominations, then, other religions, then non-religion/atheists. Which I think is fairly outrageous, but anyway.
I know differences between catholic and protestant religions are underplayed, and many practising both religions ignore some of the rules, but they do have significant differences in aspects of what they believe so its not really possible to follow both. While I think its lovely your children will be open to both faiths, and in practise they can learn about and feel part of both communities, you will have to choose one dominant one for purposes of christening and education.