To some extent, I'd say the answer is yes. Pigs and humans can share a number of viral and bacterial infections and diseases. Therefore it may be somewhat more likely that you could contract a disease from pigs that you wouldn't get from, say, birds.
However, the most like reason culprit for various cultures' aversion to pork is probably trichinella spiralis
, which is a species of roundworm which can be found in undercooked pork. The larvae of these parasites can cause trichinosis, a potentially deadly disease. This particular parasite does not rely on the great genetic similarities between humans and pigs, however. It just as readily infects various rodents, bears, and other wild game.
Nevertheless, the fact that eating undercooked pork could kill you (especially thousands of years ago) strikes me as a compelling reason to ban its consumption altogether. And what better way to do that than to incorporate that rule into the tenets of your religion!
By the way, I don't mean to offend anyone with that comment. If your religion bans pork consumption and you believe that rule is divine in origin, you can just pretend I'm only talking about all the OTHER religions that also ban it.