Does a giyur l'chumrah keep his/her family ties and tribal affiliation? I.e. does a Kohen's son who undergoes a giyur l'chumrah continue to be a kohen and that man's son, or does he shift to Yisrael and ben Abraham v'Sarah?
Giyur lechumra by definition, is a giyur that may or may not be necessary. It is performed when someone's lineage is questionable, or if they underwent a questionable giyur in the past.
As such, the giyur does not change any past status, but rather takes away any doubt as to their jewish status. In your example of the son of a kohen, he was actually only a safek kohen to begin with, because it was not clear that he was even jewish! So he will remain a safek kohen, and all ramifications will apply as before his giyur lechumra.