Is compatibility personality-based (you have it or don't) or skill-based (you develop it)?

Both. Personality provides the starting point: anxious-attached people struggle more with security than secure-attached people. But skills dramatically matter: even anxious-attached people can develop secure behavior through awareness and practice. Personality is potential; behavior is choice.

Can you become compatible through effort?

If the incompatibility is behavioral (poor communication, unresolved conflict), yes, absolutely. Both people developing skills makes huge difference. If incompatibility is values-based (one wants kids; one doesn't), no amount of effort fixes it. Know which kind you're facing.

Should I try to change my partner to be more compatible?

No. You can support their growth if they want it, but trying to change someone is controlling and usually fails. Instead, evaluate: can I accept them as they are? Can they accept me? If both yes, you're compatible enough. If either no, you might not be meant to be together.