The amount of radiation from regular flying probably has no risk as well.

Aircrew have higher risk for certain types of cancers, but it's not clear if radiation plays a role in that. And of course they spend way more time on air than any regular airline customer.

https://www.cdc.gov/niosh/topics/aircrew/cancer.html

Well and as anything radiation it is always a question of how you come into touch with it.

E.g. getting hit by radiation briefly from the outside is different than having a radiation source inside your body for potentially ever (heavy metals are known to do this).

I am not sure how it is with mushrooms, but as someone who grew up in the Austrian South with similar laws regulating mushrooms I think this was the reasoning. So not that the radiation dose of one of those mushrooms would be problematic, but that it contains radiating particles that happily enter, but not so happily exit your body.

And what is the alternative explanation for the higher risk?

Contamination and aggregation via ingestion of cs137 is very different to exposure to radiation. So can one really compare them like that, is the scale aligned?

Yes, that's the intent of the unit: https://en.m.wikipedia.org/wiki/Sievert