Arduino Mega has 16 analog inputs. They could work with simple thermistors (which you have to calibrate yourself). If you need 20, either add another Arduino or switch to more expensive digital thermosensors, which might have some adressable serial communication with a single Arduino Uno (or any Arduino).
Nothing works wirelessly. It's only a question about how long wire link you can accept and what are you willing to connect to that wire. I guess that rather than having just one matchbox size module (placed in the fridge) which takes care of the connection to your phone, you could have the single sensor inside the fridge, some wires coming out, connected to a microcontroller. And from that microcontroller it would be wireless to either Internet or the GSM net, through a suitable module.
Or if the microcontroller would be a MKRFOX1200, it would be connected to the SigFox network without external modules. One MKRFOX1200 has 7 analog inputs, so three of them would cover your 20 fridges.