Not to get all touchy-feely here ...
A Government-issued license for anything constitutes official approval of that person's status as a regular member of society. Illegal aliens are fugitives from the law, first, last and always. To grant them documents can give a belligerent sense that "I'm just as good as anybody else; you can't deny me my rights."
All well and good: we never said you weren't a good person. Our laws don't address that. We've said, until now, that you broke our laws and violated our customs by your own choice with your means of getting here, and that every moment of your presence here breaks our laws and flouts the philosophy that guides our country. Nobody (here at THR) wants to deny anybody their fundamental human rights. We're talking about the particular rights that belong to citizens and to people who have shown their honorable intentions by coming here in accordance with our laws. If we give you a Government document that's used almost everywhere as identification, we're implying to you that none of our other laws and customs really matter either, and you're welcome to replace them with your own. The problem is that ours have been shown to work better than yours; otherwise, why did you take the risks involved in coming here illegally?
Illegal aliens here are other countries' citizens, subjects or nationals. It is the business of those countries to deal with their IDs, professional licenses, driving privileges, home and school loans, and other such matters, in whatever manner they may choose to do so. We'll give some folks visitor's passes if we want to, but we are under NO obligation to allow anybody else's people to come here, for any reason at all.
The only place that buying guns comes into it is that, if they're buying from a dealer, they're identifying themselves in the process. I'll say it again: I have no problem with illegals spending money they've accumulated while they've been illegally here. Weapons, food, rent (no home loans, please). We'll even pay to ship their guns home (along with their other property) at the same time as we buy their bus (or plane, unfortunately - $$$) ticket outta here.