No, I would say it does not. The basic Nano is just a Nano. There is a later version called the "Nano Every" with a somewhat different processor which I gather does have more memory.
Oh, hang on now! Maybe you are referring to a downgraded version using the ATmega168 instead of the ATmega328. Yes, I think there are some vendors still selling that rubbish; you clearly would not want to buy those unless you had a very basic application for which that would be sufficient and the downgraded version was substantially cheaper but that is generally unlikely.
Most of us had just forgotten that version ever existed! 
Well, you are thoroughly confused then. The UNO is nothing more than a Nano with two fewer analog inputs, a basically useless "Barrel jack" feeding "Vin" and a different USB interface, on a board which does not fit breadboards and has pin sockets instead of pins.
The processor on the UNO is the same as the Nano, you could fit it with an ATmega168 but that would be pretty pointless. Actually, I think you could even fit an ATmega88 or ATmega8! 