Yubikey implements the FIDO open authentication standard. Yubico, the vendor that produces the Yubikey, is a co-author of the standard.

Read this for some background.[1]

[1] https://www.yubico.com/about/background/fido/

Yubikeys support all the different standards. Together with the Yubico Authenticator App you can use TOTP, to replace your Google Authenticator. It supports HOTP but that is used less often. Yubico itself defined a something new that is just a addition to HOTP (HOTP + Identity) and some services implement this (for example, LastPass Password Manager).

In some ways they are in technical competition but the company is making money regardless because they just implement both.

Yubikey sells U2F keys, yes. They also have their own protocol that they first started with before U2F existed, but they have been part of U2F for a while.