The current Scone stone is carved from local Perth stone. Even if that is a replacement for an original it probably wasnt the Irish stone. The gaelic kings were croned at Dunadd using the king stone petrosomatoglyph there. When Kenneth Mac Alpin was crowned King of the Picts too, he had to go to the picts king stone to do so, and it's this one that became the stone of destiny kept at scone.

The Lia Fail is the one that screamed, though. Whole other story.
As kingdoms were amalgamated, so the legends about the king stones were too. Many kingdoms become one kingdom, many king stones became one, and all the lore gravitated to that one.