Why does OpenSSL differentiate between PSS and non-PSS for private key generation?
If you reuse the same key material for different algorithms, you rely not on the security of any one algorithm individually, but on the security of the composition of the two algorithms simultaneously.

For a particularly egregious example, if you use the same RSA public key for RSASSA-PKCS1-v1_5 and for HMAC-SHA256, the results might be entertaining.

It may turn out that while we have high confidence in the security of RSASSA-PSS assuming the RSA problem is hard, and reasonable confidence in the security of RSASSA-PKCS1-v1_5 because after decades nobody's made progress on it, the combination of the two with the same key material could have catastrophic results.

The theory behind these tags is presumably that associating such metadata with the key may serve to reduce the danger of accidentally reusing the same key material for different cryptosystems.