You can still do this. It's a few more steps than you used previously.
In the message thread, tap on the picture of the person who sent the message. That reveals a sub-menu. Tap on the info icon. That reveals a comprehensive display, including all documents sent by them. You can locate the file in the documents list and proceed as usual.
The how-to I learnt from is at How to Save Audio Attachments from Messages on iPhone & iPad