JB has the setting as just after the breaking up of the unnumbered sons, So even if your marine was a part of a formal chapter they wouldn't have been in long enough to absorb any of its culture. Probably would have just found the experience odd if anything.  

The greyshields largely identified themselves by their primarch lineage as they didn't have chapters to call home. So your character would also know precisely who they are descended from as that is likely the basis for their identity at least when part of the greyshields. 


Though I am not the GM