Satoshi is not even a prince. Firstly, he is a son of a commoner.
So? Being the son of a commoner doesn't mean you're not a prince, if your other parent is the King.
Secondly, he is by no means heir to Arturia too, because among Celtic people approval of vassals was more important than the blood. That's why Gawain is the approved heir, and Mordred is not (who by blood has better claim than Satoshi).
/history lecture over/
Yeah, and Celtic Kings didn't wear plate armour, live in massive stone castles or follow the ideals of chivalry either. I don't think you can use their real-life customs to say much about the legend of King Arthur....
As for Mordred, I believe that Saber refused to acknowledge her as her child for precisely that reason. If her true descent was known then she would be the rightful heir.
But, regardless, it doesn't really matter if he is her true heir, because the kingdom he would be heir to is long-gone. Still, I would imagine it is mostly Saber's decision, and Saber certainly sees Toshi as a prince.