It is a fair bit more nuanced than a few posters here have stated.
Coutts has a financial bar to be able to bank with them. If you fall below that, they assess whether keeping you as a customer is in the wider commercial interests of the bank.
Farage was below their requirements to bank with them, they convened a board to see whether it made sense to keep him as a customer. Among financial and commercial decisions, especially in the current economic climate, it was noted that some of his very public opinions - and wider affiliations - could put the bank at both commercial and reputation risk.
This was not the main factor for him having his account closed, it was a factor in determining there were not sufficient mitigating factors to keep his account open when he did not make the financial bar.to stay with them.
This information is also totally separate to Farage and others discussing additional scrutiny when making major financial transactions (either applying for loans, or making very large purchases) because they are considered politically exposed people (PEPs). PEPs face additional scrutiny under anti money laundering rules. This extra scrutiny could highlight legal issues, or it could just highlight their finances are a lot shakier than a lender wants to deal with in the current economic climate. A PEP is not always a politician, nor are all politicians PEPs. It means an individual is at higher risk from bribery - especially from foreign actors - due to access and roles they either currently hold or have held. Lots of checks have been substantially increased on PEPs following Russian sanctions.