Hi,
First of all, an obligatory thank you for the mod message. Although small in scope, I find it to be a major improvement in visuals. I really like it when I run into such small gems that are designed well, and for one purpose only :)
Now onto the actual "issue". I have not filed it as a bug, because I think it's probably not on the side of Industrial Display Plates, but perhaps someone will find this tidbit of information useful.
I am using Industrial Display Plates with a bunch of other mods, and was a bit perplexed that copying/pasting and blueprinting the plates did not seem to copy the content (selected icon) as well - especially since you mentioned that it does support blueprinting explicitly in the description of the mod.
After reducing the number of mods, I finally narrowed the issue down to having the Attach Notes (https://mods.factorio.com/mod/attach-notes) and Industrial Display Plates enabled together.
So, when having both of the mods enabled, the display plates content is not copied via blueprints or via copy/pasting. What I find interesting is that the attach notes itself seems not to be copiable/blueprintable on its own, and this has been reported already for that mod under https://mods.factorio.com/mod/attach-notes/discussion/5ea5aeff7ed351000f257453 . This makes me think that most likely Attach Notes might be at fault in some way.
Best regards,
Branko