I agree that there’s no good reason to believe the domain provider here. On the other hand, Google’s ranking algorithms are secret and involve many different factors. It’s not impossible that Google could, for example, estimate the cost of the hosting infrastructure that particular sites are using, and rank more expensive sites higher than cheaper sites (or more plausibly, rank things that more professional or sophisticated sites usually do higher).
Anti-spam systems sometimes do something like this, and Google is surely more sophisticated than they are.
Google hasn’t made any statement about this, and so the domain provider’s claim is probably speculative (and spurious). What we’ve heard from Google is that sites with HTTPS will be ranked higher than sites with only HTTP, and Google didn’t indicate that the certificate authority mattered in this ranking.
Since Google is always adjusting its secret ranking algorithms, it may not be a good idea in the long term to believe someone who has a financial incentive in selling you something that’s supposedly good for SEO. (They may not know whether what they say is true, and can’t guarantee it will remain true!) But that doesn’t mean that the claim is impossible.
Edit: I should emphasize that although search engines might theoretically decide to consider more-expensively-hosted sites more authoritative, I don’t have any evidence that Google is currently doing this and I don’t think that certificate resellers or hosters have any evidence either.