Does power give a like-for-like comparison between different brands of stand mixer?

No, the power your mixer consumes is rather meaningless, and I would never use it in a buying decision. Maybe it is in principle possible to construct a bad mixer whose main problem is being underpowered, but since mixer wattage has long fallen victim to Goodheart's law, you can safely assume that everything on the market has sufficient wattage, and if a given model performs badly, it is not due to lack of wattage. Remember, high power consumption is no compensation for shoddy engineering.

I won't go deep into what you can use for your buying decision instead, since we have a special question for that (linked in the comments). But to hint at it, there is no number available to the customer which would allow for 1:1 comparisons. In fact, not even the numbers known to the producers would allow for such simple comparisons. So don't chase phantoms looking for a number - any kind of number - that tells you how "good" the mixer is.

In my opinion, the best way to compare mixers is to compare their customer reviews on Amazon. Even allowing for the flaws in that approach, where there are many reviews for each brand the flaws average out, and it will reflect a wide range of use/abuse by a wide range of people.

Consumer Reports (paywall) reviewed mixers in the past (last update Sept 2016). FWIW, they liked the KitchenAid Pro 6500 (list $550) the best, but of course they can't review everything. https://www.consumerreports.org/products/stand-mixer/ratings-overview/

Again, FWIW, I have read that old-school KitchenAids (like mine from 1995) were made under contract by Hobart - which was the leading brand for restaurants. Apparently the KitchenAid brand was sold and it's now manufactured elsewhere. Some people feel the quality has declined significantly since the good old days, as with everything else ;)