So you're now required to share your tax documents with the local utility company, and they will charge you a monthly fee just for earning more money?

No, the plan was changed to have a fixed monthly fee in addition to the cost per kWh for all, but lower income households will have a lower fixed monthly fee.

How do they distinguish those lower-income households without referring to tax documents?

I have no idea about this particular program, but similar programs already exist for other utilities, you can look at how they do it (open the "What information do I need to provide..." section): https://www.att.com/internet/access/

Previous proposals would have required everyone to provide income data to their utility which the utility would verify “somehow” (never fully articulated AFAIK)

The latest proposal (which has been accepted by the CPUC) has fixed charges applied for everyone (much lower than previously proposed fixed charges) with discounts available based on existing low-income programs ( https://www.pge.com/en/account/billing-and-assistance/financ... )

So yes in order to get the discount low-income customers need to declare income and may need to verify that income with tax documents (depending on how much fraud these program plan to tolerate) but other customers do not need to provide income information.