When people in the US talk about a classist system, it's more of the structure than the cost. There are state IDs that are inexpensive in most states, but the Motor Vehicles branch (or whatever the state calls their department) that issues the ID are frequently difficult to travel to in lower income areas. Occasionally, the branches are understaffed and information about what can be used as proof of ID isn't communicated clearly. Combine this with the US's poor labor laws, and it means that it's risky for someone to take off work, spend significant time getting to a DMV branch, waiting in line a long time because of understaffing, then be told they need more proof of ID and to do it all over again.

It's frustrating for me and I have a driver's license and a white-collar job so I can drive myself to the DMV easily and not worry about losing my job or losing a half-day's pay.

It's still kind of messy also because proof of birth by birth certificate wasn't required until relatively recently. My grandad was never issued a birth certificate. As far as the government is concerned, he didn't exist until he joined the army. We all have to take his word on when he was born and his name. He told it to the army and had no legal proof before then. So my state establishes identification without using birth certificates, which takes more paperwork and complications.