A Fork is a GitHub operation, not a Git one.
You can use the GitHub CLI gh fork
command to make a fork.
Example:
# Create a fork for another repository.
~/Projects$ gh repo fork cli/cli
- Forking cli/cli...
✓ Created fork cli/cli
? Would you like to clone the fork? Yes
Cloning into 'cli'...
✓ Cloned fork
~/Projects$ cd cli
~/Projects/cli$
That way, you don't have to clone any repository: the forked repository is created and then cloned for you.
This is different from creating a branch inside your own repository.
See also gh repo fork -- fork-name=xxx
with gh 2.5.0 (Feb. 2022)