No, individuals listed as Company Contacts (employees) do not automatically get assigned to a membership. The company must have a membership and membership level that allows for group memberships. Those individuals who will be participating in membership activities must also be listed as part of the membership. This can be done at the time of the memberships registration on the membership page. In the example below, the Company is the membership owner and is identified as the Primary member. The employees at that company are added as Secondary members.
If additional group members need to be added later, those individuals can be added later via the +New > Member feature available on the Memberships grid in the Memberships module. You can learn more about adding new members to an existing group membership, see this article "How do I add another constituent to an existing group membership?"