The IRS does not care if you are legal to work or not, and they won't inform on you to ICE or anyone else, but you can still get in trouble if you don't report your honest wages and pay the correct income tax.
In the situation where you have a W-2 that represents your wages, but reported in the name and SSN of another person, and assuming the employer can't or won't make the correction, I would suggest using form 4852, "substitute W-2" for a missing W-2. You can get turbotax to make this form by entering your wages and other information under your correct SSN in the program, then check the box for special circumstances for "I did not get a W-2."
I say this because, when the IRS comes looking for "Mr. Smith" who failed to report this W-2, Mr. Smith will declare themselves to be a victim of identity theft and never worked for your employer. You never really got an SSN under your own name, so in a sense, your W-2 is still "missing."
I am assuming here that you no longer work for employer #1 who issued the W-2 under the wrong name, and any job you now have is under your right name and SSN. If the incorrect W-2 was from a job you still have, then the employer really needs to correct things. I would have to think further about it.
