The issue you have linked to in that DW article is unconnected to Brexit entirely, and concerns only the Tax obligations of US Citizens resident overseas.
US Citizens are subject to domestic US taxation on their earnings made outside of the USA, even whilst residing outside of the USA, and they are required to submit tax returns to the US IRS (Internal Revenue Service) every year.
Only two countries tax their citizens like this; the USA and Eritrea.
A US Citizen living and working in the UK, earning money only in the UK (for example) is subject to US taxation as well as UK taxation.
However, to avoid the 'double taxation' issue the US Internal Revenue Service has certain financial exemption levels for each country. For the UK it is currently circa US $100,000.
If the US citizen earns up to the limit they do not pay anything to the IRS. Over the limit then the portion over is subject. The US citizen is, of course, able to take advantage of the tax deductable items to reduce their US tax liability.
The issue is a 'hot' at the moment because the US Govt require all banks in all other countries to submit financial information back to the US IRS on customers that the bank knows to be a US Citizen! If the banks don't do it the US Govt issues fines upon the banks' US operation. So the banks are chasing up their US Citizen customers to provide details, and if they don't, they close the account to ensure they don't get a fine from the US.
Trouble is there are more than a few 'Accidental Americans' in the world.
'Accidentals' are people who were born in the USA, thus had US Citizenship imposed on them from birth, but after birth resided and worked out of the USA. Typically this happens to people whose parents are non US Citizens but are in the USA at the time of the childs birth.
Many 'Accidentals' don't have a Social Security number (it wasn't until 1987 that the US issued one at birth). Others are unaware of their Social Security number, many have never held a US passport, and often the 'Accidental' doesn't realise that they must file tax returns to the US Internal Revenue Service each year.
So, if you were born to Non US citizen parents in the USA prior to 1987, never worked or lived in the USA and all of a sudden your bank in the UK tells you they are shutting your UK bank account because you haven't given your UK Bank your US Social Security number (that the US Govt never issued you when you were born and you never applied for because you never worked or lived in the USA) - you can see why these people might be a little bit concerned right now!
But then again, this issue is not connected to UK citizens residing in 'Mainland EU' past Brexit Day not being able to hold a UK bank account.
Good Luck Rapido UK.