While the previous replies offered various solutions, I found that the easiest one of them is to add the username to the repository's URL
git remote add origin https://your-username@github.com/your-username/repository-name.git
In case you have already defined the repository (without adding the username) you can update it as follows,
git remote set-url origin https://your-username@github.com/your-username/respository-name.git
When pushing to the remote repository git won't pull the credential of any other existing repository from the credential helper, but will ask for the user's/repository's specific password defined in the URL.
As a general note, I'd avoid to use your account's password at all cost and use a personal access token instead. Log in to github and select,
Settings > Developer Settings > Personal Access Tokens > Generate new token
Just make sure to check repo when defining the token's scope and enter the token (instead of your personal password) when asked for a password.