No , princes are royalty. They outrank the nobility ( peers in the UK) and commoners. Some senior princes in the Uk of course are peers too and even took their seats in the House of Lords when they were legally entitled to do so.
Princes are royalty but they are still commoners if they are not also peers of the realm. Those that had a seat in the pre-1999 House of Lords did so not because they were princes but because they were Peers of the Realm.
Britain has three broad categories of people:
1. the monarch - HM The Queen
2. Peers of the Realm - those with substantive titles - Edinburgh, Cornwall, Cambridge, Sussex, York, Wessex, Gloucester and Kent
3. Commoners - everyone else including HRHs who are not Peers of the Realm - Camilla, Catherine, George, Charlotte, Louis, Meghan, Archie, Beatrice, Eugenie, Sophie, Louise, James, Anne, Birgitte, Katherine, Michael, Marie-Christine and Alexandra (and every other Tom, Dick and Harry as the old saying is).
I always love it when people say that Catherine made such a big jump on her wedding day - from commoner to commoner. William went from commoner to Peer of the Realm on his wedding day. He was the one who made the jump in class not Catherine. (Yes I do know she went from commoner to royal but she remained a commoner).
A commoner can stand for election or vote in an election for the House of Commons. A Peer of the Realm can't. That is what actually determines their category - not any title they may hold.
I remember when Princess Anne was asked, in the early 1970s, whether she intended on voting in a general election in the UK as she was old enough, and eligible, to do so. She replied along the lines that 'it wasn't the done thing for her as the daughter of the monarch to do so'.
Queen Victoria asked Lord Pam why she should create her younger sons Duke and he told her 'so they don't stand for election to the House of Commons and lose' and that as 'commoners, alberit royal ones, they could do so'.
It was a major reason for creating Edward VIII as Duke of Windsor - to stop him as HRH The Prince Edward - from creating his own political party and/or standing for election to the House of Commons. As a Duke he couldn't do so but as a commoner - a mere Prince of the Realm - he could.