In Spain we pay the "tasa de gestión de residuos" (residue management tax) to our city, for all recycling in general. doesn't matter if it's paper, plastics/packagings, organics (many cities now collect those separat from non-organic) and other residues. It also includes the "punto limpio", a place to take larger electronics, batteries, car fluids and other stuff you can't just throw to the bin. Old medicine is collected at pharmacies though, it's a separate thing to avoid animals or people looking from the trash from getting medication, that could be dangerous.

How is recycling free but you've got to pay for organics? Don't you just pay a tax (or pay it indirectly through income tax or VAT)? And you've got the whole returning the plastic bottles on the supermarket thing too, right?

It's mad not to keep the recycling costs at the point of recycling collection, though. Why would the EU need to be involved, when it's either a private company or local government procuring the recycling service?

That's correct. In Germany we pay "indirectly" via the end price. Meaning whoever has to pay the afforementioned EPR fee is free to add their cost to the price of the product.