I don't really understand the 'your family/ your responsibilty' as any present is given from us as a family. If your issue is that it solely your financial responsibility as you have separate accounts then make sure it is fair, if it Is because you resent them then you possibly have your own reasons for keeping presents seperate.
But it's ok to have shared responsibility if it's just that: shared. From what OP has said, her DH seems to think that it is solely her responsibilty to sort presents/cards out for his family and hers. How is that fair? If OP was doing a favour by picking presents up or something like that, fair enough, but why should she have to be the one to plan, decide, get and pay for all the presents/cards?
I think it's just easier to sort your own family out (and by family, I mean the one that was yours before you met your OH), as that way you are sharing responsibility. Plus you're more likely to knw what your family wants anyway. If I had nothing else to do I would sort it, however I work full time and have a menagerie of animals to care for, so haven't the time to OH's shopping, too. Why it's seen by some as the responsibility of women, along with washing/ironing/lunch-making etc., I have no idea. Men are not children and don't need mothering or running around after, so why should OP do it?