I don't like hearing acronyms used as words, so BAME has always grated. I can see how it started though, as catch-all jargon to be used in meetings or amongst professionals who will have numerous other only roughly accurate terms that everyone knows to be generalisations.
It becomes offensive when terms like this seep out into everyday usage and enter the vocabulary of people who are not using them in this way and the fact that they are generalisations and short-cuts is forgotten.
To use less contentious examples, I dislike being referred to as 'a WASPI'. There is no such word - WASPI is an acronym which is the name of a pressure group whose aims I do not share. I am a woman who was born in the 1950s and who feels that her pension situation is very unfair, but I do not define as 'a WASPI' for reasons beyond the remit of this thread, and I dislike hearing it used to refer to all women who have lost out on their pension. My age also makes me 'a Boomer', but I also resent the implication that I have coasted through life getting everything handed to me with no effort, that I must have a valuable house that cost me pennies, a massive pension and a large inheritance, or that I voted for Brexit and dislike foreigners. None of that is true, and I think the assumption is lazy and offensive.
If people who think that others 'take offence too easily' stop to think about stereotypes that are applied to them in ways they dislike, maybe they would get closer to understanding why a lot of people don't like being called BAME.