Every organization has ideological goals. That's what I'm getting at. Sometimes they are unable to recognize it because it's been normalized. What you define as "neutral" is not actually neutral. Such statements are themselves attempts at controlling what is normal/acceptable. The terms of acceptable discussion is context-dependent, but trying to set them at "politics" is contradictory, as I've explained.
To name one easy example, for-profit companies don't have ideological goals.
But just for the sake of argument I'll grant it and amend my statement: What is usually meant by "we don't do politics" is that an institution or organization will avoid picking sides on irrelevant political issues. A cat shelter doesn't need to have an official position on abortion to function, and in all likelihood having one would be counter to its purposes.