Certainly not. As a foreigner who lived in the UK for four years, I definitely only needed my passport for international travel. I used my New Zealand photo driver's license initially for ID (eg, to get into a bar), and then my UK one. For opening bank accounts and others where you sometimes require two forms, then you bring your passport.
The UK is not like Russia or Uzbekistan where police on the street can and do stop you without cause and demand ID (in my case, six times in one day in Tashkent, Uzbekistan). And in the event they DO need your ID, a driver's license generally suffices - or they can go with you to your place of residence to get your passport if you for some reason got into serious strife.
I can only think of a handful of cases that I needed my passport. To open a bank account, to prove my visa status for starting a new job, for renting a flat / staying in a hostel, and you definitely need it for hiring a car. I remember this one as I tried to argue it, but had to eventually go back home and get my passport. I wasn't pleased.