Are HD wallets private keys any different than regular private keys
Bitcoin uses ECDSA for signing and verifying. That is, G is a curve parameter (known constant), k is a number, then the public key is kG, etc. The same mechanism is used in different script types too: Multisig scripts and other P2SH scripts. However, note that it is easier to brute-force a P2SH multisig scripts, which is why the Bech32 format P2SH scripts have longer hashes. Similarly, P2SH scripts might not use signature validation at all.

An HD private key can be encoded in the WIF private key format, and any wallet supporting importing private keys would be able to spend its outputs. This doesn't hold true for other address types.