In the simple way you describe, it is not possible; i.e., one may not add
domain.com as an administrator.
If you only want an unpaid (no email, storage, etc.) account with
superadmin privileges to manage the domain, you can use the "Cloud Identity" service. You could also use a sub-domain, but that comes with many caveats. I've both explained below.
1. Using a free (unpaid) Cloud Identity account
Using the free-tier of the Google Cloud Identity service, it is possible to have a free administrator account within
domain.com. This account will not be licensed for any paid Google services, but may still be used to log in to the Google Admin Console, etc.
Log in to the Google Admin console with a user which has sufficient permissions to create and assign whatever admin role you require for the new user.
Add the free-tier of the "Cloud Identity" service using
[fly-out side menu] > Billing > Get more services, choose
Cloud Identity in the left column, then
Cloud Identity Free. The free "Cloud Identity" service will be added to every user. Depending on the license assignment configuration for
domain.com, the Admin console may offer help to disable automatic-licenses (which will matter for the new user you are about to create, as you do not want it to receive any licenses for paid services). There is information about automatic licensing here.
Create a new account. Ensure it has no Google licenses assigned. In the
Admin roles and privileges section of the user configuration, assign whatever roles and privileges are necessary; in this case, perhaps
superadmin. There is documentation on this, "Make a user an admin".
Optional After ensuring the new account works, remove
superadmin privileges from the other paid service accounts. Obviously, you can create as many free administrative accounts as you require.
I strongly recommend all the standard security practices for the administrator account, such as 2FA or security devices, etc.
The Google "Super administrator account best practices" article is quite helpful, and discusses organization admins and roles, discouraging super admin usage, etc.
2. Considering using a "secondary domain"
It may be possible if you add
other-domain.com as a "secondary domain" of
domain.com, but this comes with various implications and limitations. The documentation is plentiful, but not particularly clear with examples and I would worry about causing confusion for the users of each domain. I suspect the domains will not be as separate as might be prefered. The documentation on Add multiple domains or domain aliases, contains:
If you own another domain, you can add it to your Google Workspace or
Cloud Identity account. For example, you manage multiple businesses or
brands, each with their own domain. Depending on your needs, you add a
domain as a domain alias or a secondary domain.
And, in the section no "secondary domains" it also contains:
Manage separate teams of users or businesses at different domains
For example, you signed up for Google Workspace with your-company.com
(your primary domain). You manage a team that has their own domain,
other-company.com. You add other-company.com as a secondary domain to
your Google Workspace account.
Which both sound helpful with respect to dealing with multiple domains.
However, further on, that documentation also mentions "Pay for each user account", which seems to imply
your-company.com will be billed for the services used by
other-company.com. This seems to confirm it:
Important: Some information and features are linked only to your
primary domain. For example, you can't set up a separate billing
address or company logo for a secondary domain.
So, as someone simply managing a domain (as a consultant, contractor, IT support, etc.) for a business, I would stay away from secondary domains. (i.e., Do not add another company's domain as a secondary for the purpose of managing it.)
"Secondary domains" seem more about different names or brands or units for a single business.