Microsoft introduced “modern” public folders in Exchange 2013. The only modern aspect of the implementation is storing public folders in mailboxes, where they can take advantage of Exchange’s Native Data Protection functionality instead of depending on the creaky replication mechanism used since the introduction of public folders in Exchange 4.0 in 1996.
Initially, Microsoft supported the migration of old-style public folders from on-premises servers to Exchange Online. For whatever reason, customers who went through the torturous process of migrating old-style public folders to modern public folders in Exchange 2013 hit a brick wall. They could not move those modern public folders to Exchange Online, even though the folders were the same type on both side of the cloud divide.
Microsoft eventually solved the problem and introduced the ability to move modern public folders from Exchange 2013 and 2016 servers to Exchange Online in March 2017. All you need is to make sure that you run the latest cumulative updates on your on-premises servers and follow Microsoft’s directions to flow data to the cloud. The process is manual and tiresome and involves manipulation of CSV files, but it does work.
It is reasonable to ask why it took Microsoft so long to deliver this functionality. When Microsoft delivered the public folder migration tools for Exchange 2013, the goal was to move customers off old-style public folders. Later, when Exchange Online introduced support for modern public folders, the tools could accommodate migration from old-style public folders, which suited Microsoft’s strategic direction at the time.