Glancing at the referenced lawsuits[1,2], the point of contention is not that Elastic's open source code is being used. It's that a.) Elastic feels its trademark is being abused in a manner that misrepresents the relationship between Elastic and Amazon, and b.) that Open Distro incorporates code in a manner that explicitly violates Elastic's licensing.
That tweet about the partnership is a pretty damning exhibit to a trademark infringement suit.
And yeah, "Amazon Elasticsearch Service" completely fooled me for about a week until Google searches revealed enough about how elastic.co isn't just a site promoting Elasticsearch, but was a provider of instance configuration.
[1]https://images.law.com/contrib/content/uploads/documents/403...
[2]https://www.courtlistener.com/recap/gov.uscourts.cand.347725...