How do I check whatever is my photo stream. This is absolutely confusing.

See the excerpt below from Information about the My Photo Stream shutdown - Apple Support

If you don't have any photos in the My Photo Stream album (or don't have that album) then you will not lose any photos.

If your photos currently in My Photo Stream aren’t already in your library, you can save them to your device.

On your iPhone, iPad, or iPod touch

1. Open Photos and tap Albums.
2. Tap My Photo Stream > Select. 
3. Tap the photos that you want to save, then tap the Share button  > Save Image.

On your Mac

1. Open the Photos app, then open the My Photo Stream album. 
2. Select any photos you want to save that aren't currently in your photo library.
3. Drag them from the My Photo Stream album to your Library. 

It is very likely you do not have My Photo Stream. It has not been a service offered with new Apple ID's for the last few years. Also if you only have one device, then there would be no need to have it. The location of the album for the iPhone and Mac are here if you have it:

• iPhone: Open Photos and tap "Album" on the bottom. "My Photo Stream" would show as a album there.
• Mac: Open photos and in the sidebar, you would have an album named "My Photo Stream.

My Photo Stream is an old syncing service introduced by Apple in 2011 for Aperture and iPhoto (My Photo Stream – Apple Support (UK)). My Photo Stream has been storing the most recently added photos in iCloud for 30 days, so they could be downloaded on our other devices, at most 1000 photos have been stored in iCloud and only for thirty days.

My Photo Stream has been replaced by iCloud Photos in 2015, when Apple replaced iPhoto by Photos. But Apple has kept it up and running for some years, so users with older devices that could not be updated to a system version supporting Photos and iCloud Photos or Aperture and iPhoto users could continue to use it. New users with a new AppleID have been discouraged from using my Photo Stream - for new users with a new AppleID the Settings to enable My Photo Stream have been hidden, so they had to use Photos and iCloud Photos instead. Now the time is up for users with older devices or the users still clinging to Aperture and iPhoto.

If you are not seeing a My Photo Stream album in Photos on any of your devices, there will not be any recent uploads to MyPhoto Stream that will need saving.

There are other ways to "share"photos. If you have iCloud Photos turned on in both of those devices, then the photos will automatically sync between them. You can go to Settings > [username] > iCloud > Photos > and see if "Sync this Phone" and "Sync this iPad" is turned on.

Also see: Set up and use iCloud Photos - Apple Support

There is no change to the Shared Photos and those files will not be deleted.