At least in the conservative community where I grew up, the religious "mark-of-the-beast/new world order" narrative was the main anti-national ID narrative. I'd never heard the illegal immigrant angle before.
They are against voter disinfranchisement. The problem is that the IDs can be made harder for some to get than others.
See momocasa's post here: https://news.ycombinator.com/item?id=19879722
It doesn't seem to be catastrophic for New Zealand, for instance.
There's a strong argument to be made that people participating in the economy and other parts of civic life deserve to have a voice in politics. It's convenient and popular to exclude them, that doesn't make it just.
(I phrase my first paragraph the way I do because I haven't thought about it much and don't have a real clear opinion 'bout it)
On the topic of residents voting, a city near me had a policy+budget allocation vote that allowed anyone who lived there and was over 13. My opinion on that is that I think that's pretty neat at a city level, but I don't think it makes sense to allow non citizens to vote in a federal election.