How is the pushd directory stack stored?

In a related question [somebody states](https://unix.stackexchange.com/a/556729/202178) that the directory stack of the pushd command is emptied when your shell terminates. But _how_ is the stack actually stored? I use [fish](https://fishshell.com/) instead of bash and the commands work the same way. I would assume pushd (and popd) works independently of the shell you're using. Or do both shells have their own implementation?