You cannot do exactly what you are asking via the ui at this time.A clients email address is their email address and a unique identifier.
It's coupled with a name. They can open as many tickets using their email address. If Joe Doe's email address is joe.doe@email.com
he can open tickets. If his brother John accidentally types his
brothers email address he can open a ticket as him too. But the name
will still be Joe Doe.
osTicket is moving towards distinct user (client) accounts similar to
how it currently handles distinct staff accounts. Staff do not have
multiple names, neither do users. If you do not want to use one email per staff
you can always add an additional field using the dynamic forms. This
field could hold "Site Contact" or "Staff name" something similar and users would
have to fill it out when they open a ticket.