Yes he is. In the books he is described as having his fathers face. The Stark looks. Eddard Stark was off to war I think for a few years and even though he was married to Catyln he still shagged some girl he met. Neds a man of honor so he took him home to raise the boy as a Lord of Winterfell. Catyln however let him live there but never accepted him.
Jons surname Snow comes from the Norths customs to name bastards.