You’re right, Strumpfhosen in German (strømpebukser) are actually tights while Strömpfe are knee-length, individual clothes that can be stockings but can also be made of thick, opaque materials. I was somehow wrong in the terminology in this area regardless of language. Socken are probably the equivalent to sokk in Norwegian as you describe.
I'd agree with your dad, although I'm Norwegian, not Danish. We would use 'strømpe' about the thigh high stockings, 'knestrømpe' about the knee high ones (aka school uniform) and 'sokk' for socks (might be fine/thin ones or thick, knitted wool outer socks).
Not sure this matches 100% Danish, though...we tend to twist the language over time...