That doesn't really fit the way web-crawler-based search engines work. If the search engine is working properly then if the new site looks the same to users it

That doesn't really fit the way web-crawler-based search engines work. If the search engine is working properly then if the new site looks the same to users it ought to look the same to the search engine. If Google's crawler didn't find the new site at all then they need a better crawler. Maybe this was incompetence rather than malice on Google's part, but it was one of the two. (An exception would be if you migrate a site and mess up your robots.txt or similar to forbid the search engine doing its job. But evidently that didn't happen here because the fix was at Google's end.)