If I were to use Suikihime sensei, or Oath of fealty to give a non-phoenix personality my clan alignment, when it dies would I be able to use my stronghold to then recruit the non-phoenix personality that was given my clan alignment permanently?
Permanently Giving a personality your clan alignment and The New Phoenix Stronghold
A topic from AEG's L5R CCG Rules forum, copied for Oracle of the Void from a copy of the forum saved in October 2015, so pages it didn't save are missing. Players' names are left out; the rules officials are named. Rulings are as they stood on these dates; later rulings, the rulebooks and the Rules Archive may differ.
If I were to use Suikihime sensei, or Oath of fealty to give a non-phoenix personality my clan alignment, when it dies would I be able to use my stronghold to then recruit the non-phoenix personality that was given my clan alignment permanently?
Yes. Merely being out of play does not end the effect: the Personality has to be face-down and out-of-play (typically, in a province or deck) - see Card Memory Rule.