But still, wouldn't you need some valid status to get one of those? I know from friends that had to get it for their child that they had to prove status if I remember correctly.

You do not need to prove any kind of status other than you are a foreigner and this is your identity. A foreign voter registration card, a foreign national ID card, or a USCIS-issued photo ID card can all be valid. It doesn't inherently mean you have the right to stay or work in the US.

That's a part of why there was a lot of hubub about ICE searching the IRS's databases for potential targets. There are a lot of people who are probably working in the US paying income taxes without authorization to stay or work in the US. The IRS generally doesn't care about your immigration status, it just cares about collecting taxes.