No.

The AMP HTML Specification and AMP-related features offered by Google Search are separate things (in the same way as Schema.org and Google Rich Snippets are separate things). If you want to have a conforming AMP page, it only matters what the linked specification says. As you noted, AMP does not require structured data.

The notice quoted from Google Search Console says that structured data is required for getting AMP-related features in Google Search, not for conforming to the AMP HTML spec. If you don’t provide certain structured data, your page doesn’t get the feature in Google Search; but nothing else happens.

One example are the Top Stories with AMP. If you, for example, don’t provide Schema.org’s image property in your AMP HTML page, your page is not eligible for getting displayed in the Top Stories carousel.

No.

The AMP HTML Specification and AMP-related features offered by Google Search are separate things (in the same way as Schema.org and Google Rich Snippets are separate things). If you want to have a conforming AMP page, it only matters what the linked specification says. As you noted, AMP does not require structured data.

The notice quoted from Google Search Console says that structured data is required for getting AMP-related features in Google Search, not for conforming to the AMP HTML spec. If you don’t provide certain structured data, your page doesn’t get the feature in Google Search; but nothing else happens.

One example are the Top Stories with AMP. If you, for example, don’t provide Schema.org’s image property in your AMP HTML page, your page is not eligible for getting displayed in the Top Stories carousel.