SIG parts interchange depends on both model and version.
The German P220 9mm and the German P226 have the same barrel. All of mine swapped without issue, and up until recently had two P220s and two P226s. My P220s are the side mag release pistols, though I believe the bottom mag release pistols will also interchange. With German guns of the same vintage.
My P220s also take modern Storm Lake P226 barrels without issue.
However, a new barrel from the latest "Caliber Xchange" kit for a P226 in 9mm wouldn't fit. Just to look at them, they are very similar. I didn't have the time to sit with calipers and micrometers and figure out what's different.
So our two posters' replies above are right, but so is mine. "It depends".
I also learned today that the takedown lever is compatible between P220, P225, P226, P228, P229 ... I'm sure somewhere there's a list of parts that fit between models and versions.