Where to store my Git personal access token?

Is it necessary to store the personal access token somewhere locally on the machine after generating it in GitHub?

If yes, is there any preferred way where it could be stored?

Half the point of passwords is that (ideally) you memorize them and the system hashes them, so therefore they're never stored anywhere in plain text.
Yet GitHub's personal access token system seems to basically force you to store the token in plain text?

First, a PAT (Personal Access Token) is not a simple password, but an equivalent that:

• you can generate multiple time (for instance, one per machine from which you need to access GitHub repository)
• you can revoke at any time (from the GitHub web interface), which makes that PAT obsolete, even if it lingers around on one of those machines.

That differs from your password, which is unique to your account, and cannot be easily changed without having to also modify it everywhere you happen to use it.

---

Since a PAT can be used in place of a password when performing Git operations over HTTPS with Git on the command line or the API, you can use a git credential helper to cache it securely.
On Windows, for instance, that would use the Windows Credential Manager, through the GCM -- Git Credential Manager -- for Windows, Mac or Linux:

git config --global credential.helper manager-core
# Git 2.39+
git config --global credential.helper manager

(manager-core is being replaced by/renamed as manager for Git 2.39+, Q4 2022)

The first time you are pushing to a repo, a popup will ask for your credentials: username and your PAT.
The next time, it won't ask, and reuse directly that PAT, which remains stored securely in your Credential Manager.

A similar idea applies for Mac with the OSX keychain, and Linux with the GNOME Keyring (in 2021, it would need a DBus session and libsecret), but in 2021, GCM-Core covers those use cases.
The idea remains: store the PAT in an encrypted credentials store.

---

As mentioned above, the more modern solution (Q4 2020) is Microsoft Git-Credential-Manager-Core, or, Q4 2022, Microsoft Git-Credential-Manager

git config --global credential.helper manager-core
# Git 2.39+:
git config --global credential.helper manager

---

Before Git 2.39 (Q4 2022), for Linux:

You need for that to install git-credential-manager-core, downloading its latest release, like gcmcore-linux_amd64.2.0.474.41365.deb

sudo dpkg -i <path-to-package>
git-credential-manager-core configure

Although, with GCM (Git-Credential-Manager-Core) on Linux, as noted by Mekky Mayata in the comments, you need to define a git config --global credential.credentialStore first.

See "Credential stores on Linux":

There are four options for storing credentials that Git Credential Manager (GCM) manages on Linux platforms:

• freedesktop.org Secret Service API
• GPG/pass compatible files
• Git's built-in credential cache
• Plaintext files

By default, GCM comes not configured.
You can select which credential store to use by setting the GCM_CREDENTIAL_STORE environment variable, or the credential.credentialStore Git configuration setting.

As noted by agent18 in the comments, using git-credential-libsecret after installing libsecret-1-0 and libsecret-1-dev is a good first step.
But, again, that should be now wrapped by credential-manager-core (before Git 2.39).