How to change the installation user's group name

"Migrating all users to a new group does not help, as the first user, who is also an administrator, still sees the groupname by default equal to his first name."

The migrate that user too. There's nothing special about the first user created in installation, nor is there anything special about administrators, with respect to primary group membership.

The only think I can think of that might make it seem otherwise is if you're neglecting to log that user out and back in, before using the account to create a file and see if the file's group owner is the new group.

You can perform the migration however you like. I'd probably use users-admin (System > Administration > Users and Groups), because it's graphical and easy and fast. To do it in that utility, click on the username, and then click the Advanced Settings button near the lower-right corner of the window. When the Change Advanced User Settings window comes up, click the Advanced tab and change the "Main Group" to whatever you want.

To answer the other part of your question, though it might not be relevant to the solution you implement: You can remove groups and change the ID number of groups. (The group owner of a file is stored as the group ID number of the file. If you change what group has that ID number, you change the group owner of the file. Therefore, changing groups' group ID numbers enables you to accomplish most of what you could accomplish by changing the names of groups.) To edit groups in users-admin, click the Manage Groups button. To delete a group, select it and click Delete. To change a group's ID number, select the group and click Properties.

After you change a group's group ID, if you want the files that formerly had that group as their owner to still have that group as their owner, you'll have to chgrp them again (see below).

If you really, really wanted to change a group's name, you could make all the necessary substitutions in /etc/groups and /etc/passwd. But there is really no need to do this.

On the other hand, if the issue you're experiencing (or an additional issue you're experiencing) is that the files formerly created by users (this would happen for all users, not just the first one) still have their old group ownerships, then the solution is simply to change their group ownerships. One way to do this is by running:

sudo chgrp -R bond ~james
sudo chgrp -R bond ~samantha
# and so forth

"Before I get some inane questions on why would this be ever required..."

There are plenty of reasons why someone might want all or most human users on a system to have the same primary group, and many other Unix-like operating systems default to this behavior (naming that group "staff"). However, the specific reason why your clients want this seems quite silly. But there shouldn't be any harm in it...and as the saying goes, whatever floats their boat...