I always go for the realistic style in games, and was wondering, could someone who is very skills in using an axe and a shield, be skilled in using a bow and arrow as well?

Of course.

However, I have an issue in which I don't like using shields while having bows. I mean where would either go?

Same with Two Handed and Archery.

Of course they can. There's no law in nature that says one can't have more than one skillset.

Just look at modern day soldiers, they recieve traning in h2h combat as well as long range engagements.

I think as long as a person could personally carry them, anything was possible. Everyone has their limits, but I doubt it was uncommon for someone to be equipped with what you have. Viking Raiders who doubled as hunters might have used similar equipment.

Yeah I have that issue with two-handers and bows,I don't see the problem with shields though.

If you would've done it in real life, why not?