Yes, Project/Group Access tokens are available in all tiers (Free and paid) for self-hosted instance.
If you are on self-hosted and cannot create any token, it was most likely forbidden by your GitLab administrators or you do not have permissions to do it. Either way, you need to reach out to your GitLab admins.
I try to use tortoisegit to git clone my group project and it show the sign in with gitlab.
And then I can git clone my project without token ( GitLab.com and Free).
Do you have the tourial (command) for login gitlab without tortoisegit ?
Keep in mind that access tokens can easily be used at all available levels when it comes to your project or group, taking into account the installation we hosted ourselves. I also recommend that you use tortoisegit.