surfhb nailed it. Your SO would not have both a 401K and a 403B available to him, but rather one or the other at any given employer, and they're so close to the same thing that, for someone of your SO's current financial literacy level, treat one as a typo for the other.
If he continues to insist on not investing for retirement, you really have only a few choices:
1. Encourage him to become more financially literate and invest for your future.
2. Choose employment at a company that still offers defined benefit plans (a traditional pension).
3. Agree to give up control over the savings/investments to you (or to someone qualified).
4. Be willing to accept that you are not likely to join the path to FIRE.
5. Change horses.