"We don't store your biometric data, just a number in one-to-one correspondence with it that anyone with access to our algorithm could reproduce..."
Converting an image to a template requires similar images to result in similar templates. That's usually the opposite of what you want from a hash. There is a concept of "biometric hashing" that would accomplish the one-way claim, but it's theoretically hard and as far as I know none have hit the accuracy levels to make the biometric part feasible for a large scale systems.[1]
The real claim biometrics vendors tend to make if you push them is that you can't reconstruct a photo-realistic image of the face / finger / iris from the template. Which is generally not a true claim either, but at least would require some work with generative tech. And a photorealistic image that generates the same template is likely to look a lot like the actual person.
Source - 10 years of consulting in biometrics for government, academia, and industry, including spoofing and vulnerability assessments. Most people in the industry don't know what they are talking about and just repeat marketing claims.
[1]https://www.cse.unr.edu/~bebis/CS790Q/PaperPresentations/Bio...
Biometrics do freak people out more than tracking cookies do, largely because they represent your actual body (e.g. this article refers to an IR photograph of your iris as an "eyeball scan" because it gives the reader a more visceral reaction - "eyeball" makes you think of something wet and biological, and "scan" sounds invasive and like it involves lasers or something.).
But they can solve some serious technology difficulties in places where other identity infrastructure isn't in place, and often the concerns are a bit philosophical compared to the tangible benefits. As with all privacy-invasive tech, the issue is exposure to rare but nasty failure modes, and biometrics has the difficulty of being non-revokable.