Even assuming an eschatological evangelical worldview, it isn't immediately clear to me what distinguishes IDs from any other engagement in society.
This, obviously, creates a problem for Christians who are commanded to love God and to love others.
If it was just an identifier that held no other meaning, then it's fine, but it's called the "mark of the beast" and it's described as a literal badge of association with this creature. It's not just a bank account number.
Hope that helps. If not, just imagine how odious it would be, especially to Democrats, if Republicans made it a requirement to wear your MAGA hat to buy food.
I'm agnostic and always found these sorts of religious doomsday proclamations to be exaggerations, but after seeing what the masses have openly embraced over the last 2 years I know I was wrong. It hasn't made me religious, but I can definitely see their concern for being forced into any number of situations in order to fully participate in society.
It's not worth it.
This is a parody by non-Christians of what most of them actually believe. For most of them the concern is over microchips, implants, or something physically on their person, not an ID Card, and even there is not uniform concern over it. Revelations itself says that it will be on their right hand or forehead, which no ID Card is. One could argue that it is a foreshadowing of things to come, if not itself it.
As for where the concern comes from, it's not the ID itself, as much as the implication that in the future, people will be required to commit apostasy in order to get an ID, or the chip, or whatever have you. Thus, they are opposed, so as to prevent such a lockout.