There is no law which stops people from renting booster seats outside airports.
I just checked a few car rental agencies and Avis, Hertz and Sixt offer such seats in some city locations, so it’s just that one company you contacted.
Those two are not the cheapest agencies so you might want to use a price comparison site such as Billiger Mietwagen. Unfortunately that site is neither available in English nor does it allow to filter by available booster seats (nor does any other site I checked).
There is also a company in Berlin I found which is specialized in renting booster seats.
Credit:stackoverflow.com‘