I've seen Energy Star logos for 30 years and never knew there was a public database, never thought to verify, and I don't think anyone else has either. The only thing Energy Star has been useful for is extracting rebates from utility companies and buying shitty dishwashers which were certain to be worse than what they were replacing.
Verification is useless if no one knows about it, or if the data isn't actionable. I have verified UL mark numbers for questionable products, but they often resolve to some Chinese ODM you've never heard of like 'Xionshang Industrial Electric Company' whose name certainly doesn't match the product label. Do you know the components haven't been swapped out since certification was achieved? Was the product actually sourced from there or counterfeit? You have no way to verify any of that.
UL issues holographic stickers but I've seen those like 10% of the time and probably just as easily faked.