You can continue using the issued client certificate even if you change the user details, such as login name and password. You do not need to issue it again.

However, if you change the status of a user to "Inactive" on the "Users & system administration" screen, the issued client certificate will become invalid for the user. If this happens, you cannot make the client certificate valid again even if you change the "Status" of the user back to "Active". You need to re-issue the client certificate.

Additionally, you also need to re-issue the client certificate to change the information in the client certificate itself.

For the steps to re-issue the client certificate, refer to the following page.
Configuring Client Certificate Authentication