Identity License is a minimalist license different from the External Identity license.
The primary difference is in the Object access granted. Identity DOES NOT provide access to Standard or Public Sector Solutions Objects, whereas External Identity provides access objects listed in it's respective documentation.
Attempts to grant access to any Standard or PSS objects, whether by Permission Set or Profile, will result in the following error:
"The user license doesn't allow the permission: ..."
There is currently no plan to expand the license object access or provide a workaround. The following is currently advised by Salesforce Engineering:
"External Identity grants access to Contact/Account objects but the user should have user license as salesforce and profile as standard user."
External Identity License Details:
003876541

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