If a photo in "Portrait" mode with blurred background was edited in 3rd party app (Photoshop) then after saving in Photoshop, "Portrait" mode disappears, background becomes normal. What I should to do to save "Portrait" mode after editing in 3rd app?

I am duplicating the Portrait mode photo before using the external editor, so I will still have the original portrait.

And then I am calling the external editor as a Photo Editing extension from Photos. It will get the photo passed with the depth effect already applied.

You could only do that if you have a third party application that understands Portrait mode - and possibly you can export the portrait mode photo "burning" the effect into the exported photo and then edit the exported photo

LN