In Beverly Cleary's book Socks, it is described how the characters often wipe the dog's paws with a good bath towel whenever he comes in with wet feet. This shows their care and the routines they have established to keep their home clean and comfortable.
The quote highlights that they did not yet have an old towel for such purposes since they had not been married long enough to have accumulated many linens. It reflects their early days of building a shared life and the small, everyday acts of love and practicality they share.