Can people skip the code on their own computer?

Yes, if you allow it. Settings → A second step → Let people skip the code on a browser they choose is off on every site. Switched on, the code screen offers Do not ask on this browser for 30 days; ticked, that browser then signs in with the password alone until the days are up. Choose the days (1 to 90) and the roles it is never offered to: administrators are ticked to start with. It is offered only over HTTPS.

It is a trade: on that browser, a copied cookie and the password are enough. The cookie is HttpOnly, Secure and kept for your site only, and a copy works once at most. A new password ends it, and so does Forget or Forget all under Remembered browsers on the person's profile, Forget every remembered browser in Settings, or wp login-security two-factor forget <user>.