Let them eat foie gras! It was only a matter of time before they started manipulating index funds too.
Vanguard is probably the most principled when it comes to passive index tracking, and they do not have an ETF that tracks the NASDAQ 100 (or any fund that focuses on a single stock exchange for some inexplicable reason).
Yes, you can. Whether or not the index makes sense for whatever one's investing goals may be is irrelevant.
For example, a fund buys an equal number of shares of every publicly listed company.
Or a fund buys securities that trade with a ticker symbol starting with the letter C.
Based on that definition, one could refer to a Cathie Wood index fund to be composed of whatever she decides to buy and a bagacrap index fund to be composed of whatever bagacrap chooses to buy.
QQQ is both arbitrary (the stock exchange has nothing to do with business operations) and capricious (they are changing the rules on a whim to serve Elon's interests, and because the arbitrary inclusion criteria somewhat forces them to).