All telephones in the U.S. were as a rule property of the local telephone company, which in North America would have been owned by AT&T (American Telephone & Telegraph Company). AT&T was founded in 1885 by, among others, Alexander Graham Bell and continued to hold a monopoly for phone service in the U.S. and ownership of all phones until about 1982. This means that the Lane's phone did not belong to them; it was merely rented from the local phone company, hence the $20 deposit for installing a second phone.
Adjusted for inflation, the $20 deposit for a second phone would equal approximately $185 in the year 2023.
The telephones used in the Lane house and the phone Martin uses in his office are all the same model, i.e. the Western Electric model 500 rotary desk phone. This model was first distributed to phone customers in 1950 and continued to be the standard until push-tone technology caught on in the late 60s, early 70s.