How do you actually do the income taxes though?

Many companies can do payroll withholding for local/school taxes just like they do with State or Federal. There are boxes on your W-2 you get when you file your taxes that show this:

Is it completely separate from your federal/state that get filed each year?

Yep, 3 forms to fill each year: Federal, State, Local/School

It seems like it would be a huge burden to be a separate thing, and it would be easier if it was somehow incorporated into a federal/state tax system where if your municipality has taxes you fill out a few spots and the money gets sent to them?

Its a pain, but so are Federal and State taxes. Municipal/school taxes can also get complicated if you work in one city with tax and live in another. Many will give you partial or full credit so you don’t end up paying full municipal income tax to two cities, but this isn’t always the case.

E.g an added section on your state/federal to add the municipalities tax id and tax rate (as it’d be variable by area)

Just like each State has their own rules, each city can have theirs so your suggestion can’t be just numbers to fill in because which numbers may change between cities. Dividend income is one example I know different cities treat/tax differently. Yes its a pain, but, its adulting.