Summarize:

It’s the blue chat bubble with a white lightning bolt in the app drawer. Use this method any time you want to save a picture from Messenger to your device. If you want all pictures in Messenger to save automatically, see Saving Photos Automatically.  It will open in full-screen mode. It’s at the top-right corner of the screen. The photo is now saved to your gallery.
Open Messenger. Select a conversation. Tap the photo you want to save. Tap ⁝. Tap Save.