Not really. Things that you 'stick with' are things that you have to put time and effort into. You might 'stick with' a relationship, or 'stick with' an old car. You need to keep the relationship going, and keep the car going to keep them working.
To me, things that stick with you are more things that you remember, or things that keep coming up. Perhaps an old habit, or a thing that you keep doing or saying, for no reason, other than it was what you did a long time ago.