Somehow people managed to vote and have bank accounts before ID cards were common.
The UK wants a National Insurance number to register to vote. https://www.gov.uk/register-to-vote
UK residents are sent a National Insurance number automatically just before their 16th birthday. https://www.gov.uk/apply-national-insurance-number
Pulling up an arbitrary UK bank, https://www.tsb.co.uk/current-accounts/faqs/identity/ shows many options to establish identity, including ones that don't require an identification card. Those under 18, for example, can use a National Insurance card or letter.
I don't know what domestic flights are like in the UK. From what I can tell, it's left to the airlines. Most (? all?) require some sort of id, and different airlines accept different forms. For example, FlyBe allows "Council issued bus pass" and "NUS card (National Union of Students)" and "Company ID card of nationally recognised company" - https://www.flybe.com/check-in/id-requirements .
In some countries (I'm thinking Australia, based on a podcast I listened to from a couple of years ago), domestic check-in is all automated and does not always require an id check. https://www.jetstar.com/au/en/help/articles/travel-identific... says "If you’re travelling with checked baggage on a domestic flight in New Zealand, you must provide ID." "Accepted ID types" include "Original or certified copy of a birth certificate", so "ID" is pretty broad.
Something like 17 states in the US don't require any form of id to vote - https://en.wikipedia.org/wiki/Voter_ID_laws_in_the_United_St... - so this isn't unique to the UK.
To point out, voter impersonation - the sort of fraud which ID cards might solve - are very rare. See https://en.wikipedia.org/wiki/Voter_impersonation_(United_St... . The vote is private, so if there was an organized system it's difficult to verify that the vote actually went the correct way. It requires a lot of people in order to sway an election, and that sort of organization has a higher chance of being detected. The list of people who voted is public, so it's also possible to track down if they actually exist, and there's the risk that a impersonator will try to impersonate someone who actually voted.
If it's important to know who someone is, then it can sometimes be solve by using a chain of trust. For example, the bank might require that a current account holder in good standing be willing to vouch for me, even if I have no id. Or, I might deposit a lot of money but have limited abilities until the bank trust me more.
Regarding the UK, see also https://en.wikipedia.org/wiki/Opinion_polls_on_the_British_n... .