Tiaras and cornets

In Knight Squad Ark, a student at Knight School in the Kingdom of Arcadia, knows that fellow knight student Chiara is actually the princess of Arcadia, who uses a magic pixie ring to disguise herself as Chiara and attend Knight School since her father the king won't let her attend it.

In episode S1:E10 "Wish I may, wish I knight" the princess appears in public wearing a tiara. Later Ark visits her bedroom while she is talking to her closet full of tiaras.

And that is part of the way that popular culture associates wearing tiaras with being a princess. But anyone who is rich enough to afford to buy a tiara can wear one to parties and other formal occasions. Princesses who wear tiaras do so because their families are wealthy enough to own tiaras.

The headgear that is associated with princesses and princes and indicates their rank is coronets.

These wikipedia articles about crowns and coronets shows the coronets associated with various ranks in various current and former European monarchies.

https://en.wikipedia.org/wiki/Coronet

Note that sometimes the coronet of a "prince" is that of a nobleman with the title of prince, and sometimes that of prince belonging to a royal family.

In most of those countries the coronets of royal princes and princesses exist only on paper, as parts of their coats of arms. But certainly royal princes and princesses could have real physical coronets made that show their ranks instead of princesses wearing tiaras that are merely decorations.