Any rabbit, guinea pig, hamster, potbellied pig, bird, lizard, snake, turtle, or tortoise that is legally allowed as personal property and that is impounded in a public or private shelter shall be held for the same period of time, under the same requirements of care, and with the same opportunities for redemption and adoption by new owners or nonprofit, as defined in Section 501(c)(3) of the Internal Revenue Code, animal rescue or adoption organizations as provided for cats and dogs. Section 17006 shall also apply to these animals. In addition to any required spay or neuter deposit, the public or private shelter, at its discretion, may assess a fee, not to exceed the standard adoption fee, for animals adopted by new owners or released to nonprofit animal rescue or adoption organizations pursuant to this section.