Yubikey is only really useful for authentication with a trusted party, and not decryption. You can technically use store a secret key on it but then its two biggest advantages are gone, namely that you can't copy the key and that it doesn't use the limited storage on the device.
The yubikey can perform a hmac using a secret (supposedly) only available to the key's internals. This is used in addition to the password, so that knowledge of the password without the key, or the key without knowledge of the password, can't be used to decrypt the database. It's kind of a half second factor (I know it's not technically correct to call it that, but I hope you get the idea).
You can absolutely copy the key, because the device has to give it up to the application during decryption. Or does the application send the encrypted file to the yubikey for it to decrypt it? In which case, that's a lot better and I'm wrong.
You use a GPG key that you then add to the yubikey. The keys can only be written or deleted off the yubikey, you can't read the secret once written. Then you can use the GPG key to either encrypt a file or sign it. Check out Pretty Good Privacy and the GnuPrivacy Guard software for more information on how that works.
I use my yubikey to encrypt files, sign my work in Git, as well as the usual password authenticator stuff. You can still use FIDO, U2F and OTP codes while using the GPG too.
Excuse my surface knowledge on this but when setting up TOTP on Yubikey you can choose to only get an OTP on touch which would be pointless if the application had access to the secret at any point. Based on that it's probably not possible to copy it.
You're talking past each other, some Yubikeys have PGP apples for asymmetric encryption (public / private keypairs), and HMAC is a symmetric single key algorithm where the yubikey sends a resulting value to the PC/phone which is part of the key derivation inputs (even though the yubikey's root key remains secret).
i use keepass to store all my passwords, the database file gets synced across my devices through Dropbox, i open it with a master password, i would like to improve this by also requiring the yubikey
i am kind of confused too as to what exactly the yubikey does in this scenario. my vague understanding is that it was somehow synchronized such that the yubikey would generate sequential random 'passwords' which would be checked against the database file (generating its own sequence in the same manner).
i think it stopped working due to some desynchronization between the yubikey and the database file.
Sync shouldn't really matter, unless you're using a hotp code as opposed to a certificate or TOTP code.
TOTP being temporal, is based on UNIX time, and a seed key. A certificate will be challenged, which will require a challenge and reply all cryptographically encrypted. It's not something that's necessarily stored in some kind of sync, or rolling codes.
I'm not familiar enough with keepass to say what it's supposed to use with the yubikey in order to work. There's a few other methods that I'm sure that keepass could leverage to perform the authentication, so I'm not entirely sure what could be the problem.