What does affiliation (for a publication) signify?
Affiliation could mean anything, or nothing at all. In most cases it doesn't mean that the organization approves or even knows of the publication beforehand. In some companies you sign a contract that lets the organization vet your public work, but not in others and not in most universities. In those companies it usually only means that they approve of the fact that you are publishing, but not necessarily of the content. There can be exceptions, however, as when you are working with sensitive information, such as trade secrets.
When I published, I listed my employer (a university) as my affiliation, but that was for identification purposes only. I probably still would, though I'm retired. I might, instead list none or independent researcher or my DBA (Doing Business As) persona.
I think that publishers want it almost entirely for identification purposes, nothing more. However, for some affiliations, you have a sort of implied aura that gives the editor a warm and fuzzy feeling.
Generally listing affiliation serves two purposes:
For readers it helps with locating authors to contact them (for questions on their work, collaboration, reprints, etc.)
The institution, be it a university, a graduate school, a hospital, etc. can list the published research wherever they need to show research activity record. This is often critical for funds and resources allocation.
Usually, the principle of academic freedom (which notably isn't about a right to skip classes as many students seem to believe) should imply that the views are of the author alone. In reality I think you can't prevent readers, especially in the general public, from inferring some sort of endorsement by the institution. Thus papers published by prestigious universities' affiliates might be seen as more "valid". Within a research field I have not seen significant evidence of this though.