Can a family member have their own login?

Yes. You have control over what their username and password are. You can set these in the Family members tab in My account.

Choose the family member from the drop down choice and they'll be a box inviting you to set a user name and password. Set these and ask your family member to login with those credentials.

Once your family member has logged in they can control/change their own user name and password or add their own email.