When a dog or cat licensed or registered in one county or city is permanently transferred to this City, the owner shall, within 30 days of such transfer, surrender the original license or registration to the City Clerk. The City Clerk shall retain the surrendered tag and, without registration fee for that calendar year, shall issue a new tag pursuant to this chapter upon completion of the required registration application. The City Clerk shall record the fact that the newly issued tag is issued as a transfer and is in lieu of the surrendered license or registration.