The Client Tracker tab is a dependant tab of Contacts. You cannot create a Client Tracker record without relating it to a Contact record. You should create a new record for each Client Tracker process of getting a dog. Successor Dogs should start a new Client Tracker record.
While you can create Client Tracker records from the Client Tracker tab, it's best to let the system create one after you create a Contact record with the Client record type. After creating the Contact record, a link to the Client Tracker record will appear in the Current Client Tracker field.