ghost touch selected it.
I'm on lollipop.
Edit : screenshots for comment below. I also checked Reset Default Apps in,the Settings menu, but there's only e-mail and messaging.
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If It Is ( Google) Drive Pdf Viewer Then :-
I don't use google so no data (of mine) to lose, and that did it.
I did manage to find the core problem, tho. Opening an image file did the same problem : ghost touch selected Default as I tapped Gallery. Of course, Gallery is right on the application list but again no default was set there or under Clear All... but I deleted the data and fixed it.
Something clicked that I thought to check ES File Explorer so I set Gallery as Default again and checked that app, ES. It didn't have a Default but I deleted that data and that also restored the Open With... menu!
So that's the problem : browsing with ES or maybe other file managers and selecting a default only applies to that manager (confirmed via browsing with the stock manager), and why no default shows in the application manager, and the solution is to delete data for either the defaulted application, if known, or the file manager.
Poor design on either ES or google's part or both.
Thanks, @Kirandeep Maan, for the help and getting this solved!