How do I make Git ask for a username and password every time I push?

If you use a Windows system, the Git username and password is stored in a control panel. You can go to Control Panel → User Accounts → Credential Manager:Generic Credentials to delete it.

It seems that users credentials are being cached.

Go to your project, open .git/config and remove the lines:

[credential]
    helper = store
    # or helper=cache

Check git-credential-store for more information.

Edit: OP mentions in the comments that this is happening in Windows, for which, the other answer is more relevant for removing the credentials from the Control Panel.