This seems to be dependent on the Android version. My old Samsung phone has Android 10, the new one has Android 12.
Both have the same Samsung Music App V 16.2.27.5 installed.
Only on the Android 12 phone I could find
Samsung Music: ... 》 Settings 》Manage Playlists 》Export My Playlists
and
Samsung Music: ... 》Settings 》Manage Playlists 》Import Playlists
Hint: Settings can be found in the three dots menu (...) on the upper right corner of the Samsung Music App.
The playlists are stored in *.m3uformat in the internal memory, directory Music. This is an editable text format.
Note: Nevertheless, I was able to transfer the playlists from the old to the new phone by using Samsung Smartswitch. Just ensure the music is on the target phone before running the Smartswitch (use TotalCommander with WiFi plugin etc to copy it). Also important: After copying, run Samsung Music App and give it the time to discover the music on your target phone. Then finally, run SmartSwitch on both phones to transfer apps + settings.