Existing organizations getting updated when new person signs up with the same org name
I am wondering if you really want the membership form to be "on behalf of" an organization.

That feature is designed to be used when the organization is to be the member, (eg for a trade association) but, because organizations can't fill in forms, a person has to sign up "on behalf of" the organization. The organization can only have one membership, so when someone else comes along to sign up 'on behalf of' the organization they are effectively renewing that membership and will be added to that organization.

It sounds like you want the individual to have the membership, but you want to know that they belong to a particular organization.  If that is the case then your memberships should be for individuals, you should not let contacts sign up "on behalf of organizations" and you should include the "employee of" field in the profile on your membership form so that you have a relationship created between the organization record and the individual's record.