That’s not how I read Donal Noye’s speech:
“No. They hate you because you act like you’re better than they are. They look at you and see a castle-bred bastard who thinks he’s a lordling.” The armorer leaned close. “You’re no lordling. Remember that. You’re a Snow, not a Stark. You’re a bastard and a bully.”
“A bully?” Jon almost choked on the word. The accusation was so unjust it took his breath away. “They were the ones who came after me. Four of them.”
“Four that you’ve humiliated in the yard. Four who are probably afraid of you. I’ve watched you fight. It’s not training with you. Put a good edge on your sword, and they’d be dead meat; you know it, I know it, they know it. You leave them nothing. You shame them. Does that make you proud?”
Jon hesitated. He did feel proud when he won. Why shouldn’t he? But the armorer was taking that away too, making it sound as if he were doing something wrong. “They’re all older than me,” he said defensively.
“Older and bigger and stronger, that’s the truth. I’ll wager your master-at-arms taught you how to fight bigger men at Winterfell, though. Who was he, some old knight?”
“Ser Rodrik Cassel,” Jon said warily. There was a trap here. He felt it closing around him.
Donal Noye leaned forward, into Jon’s face. “Now think on this, boy. None of these others have ever had a master-at-arms until Ser Alliser. Their fathers were farmers and wagonmen and poachers, smiths and miners and oars on a trading galley. What they know of fighting they learned between decks, in the alleys of Oldtown and Lannisport, in wayside brothels and taverns on the kingsroad. They may have clacked a few sticks together before they came here, but I promise you, not one in twenty was ever rich enough to own a real sword.” His look was grim. “So how do you like the taste of your victories now, Lord Snow?”
My interpretation is that Donal Noye is arguing that Jon shouldn’t think himself their superior just because he’s better than they are at swordplay, because his superiority is based on an inequality of educational resources.
As to your question, GRRM’s version of stigma against bastardy is more severe than historical parallels in medieval Europe. But absolutely, medieval societies largely preached the idea that those “gently born” were better than peasants.