Using 5 V adapter to power 12 V cordless phone
Theoretically it works - 


No it doesn't. In theory 5 V x 1 A = 5 W maximum from your PSU. Your phone needs 8 W and you'll lose another 1 or 2 W in the voltage booster. It doesn't work in theory and, as you've found, it doesn't work in practice.


  ... when I connect adapter to boost convertor input and setting potentiometer to 12 V by measuring output through multimeter - it shows 12 V.


That's the open-circuit or unloaded voltage.


  My question is the DC coming from adapter different than DC coming from battery?


Yes. The battery has a much higher current limit.


  Also I noticed that the adapter indicator light is constant when charging a basic (non smartphone) phone battery but it pulsates when connected to step up convertor and cordless powered on. This made me more confused.


Why the confusion? Check the manual for the adaptor and you'll probably find that it's indicating over-current fault.