After looking at this https://wordpress.stackexchange.com/questions/88327/importing-from-xml-is-ignoring-serialized-custom-field it seems that importing serialized meta values has issues. I tried debugging the lines with no success.
Then, following one of the suggestions, I copied the postmeta information directly from one database to the other. It worked! This at least gives me a tedious solution.
After the database copy worked, I tried to export the products on the new site to compare the exported data with the original exported data. There didn’t seem to be any differences, but this export also failed to import again. So, this is not an issue of it being a different site at all, because I have tried to export a working tabbed product from the new site, deleted the product completely, and imported the product XML and the tabs again are not imported.
Based off of the linked discussion, I am wondering whether special characters, multiple lines, URLs, or something else is causing the exported information to be regarded as invalid when WP Import attempts to import it again.