Perhaps in the very murky 'not quite feudalism' era of 867 and before, sure, and especially when it came to pagan cultures or those which were rooted in paganism (i.e. the 'Danish' or 'handfast' common law marriages of the Scandinavian kings) later on, but for most of Europe it didn't happen[...].
No king of France - even during the post-feudal age during which they wielded far greater power and often ignore Parlement - had ever legitimized a bastard and made him a prince of the Blood and part of the succession. It wasn't permitted and would never have been sanctioned.
Pierre-Charlot (1205?-1249), the bastard son of Philip Augustus and a "certain lady from Arras" (according to Guillaume le Breton), was officially legitimised in indult by pope Honorius III - sure, he was far down the line of succession and would go on to pursue an ecclesiastical career anyway, receiving the diocese of Noyon in 1240, but he was acknowledged as a legitimate son nonetheless.
From the 14th century on, the Valois monarchs of France claimed the right to legitimise bastards as a royal attribute, seen as essential as the conferment of nobility or the creation of notarial offices. There are several examples of such legitimisations in the Late Medieval era - Anthony, the Grand Bastard of Burgundy, is a very prominent one, being right at the tail-end of the CK3 timeframe.
And yeah, sure, the legitimisation of bastards was never super common. But it wasn't unheard of, nor was it restricted to cultures with Norse or Celtic influences.
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