May have been too late, but as you followed the recommended steps, it all worked out in the end.
What happened:
Some hacker/developer/corporate thief who has a z3c noticed that the backup and restore app was actually really good. He then had the idea of doing a z3 system dump to extract the app.
Once extracted he changed the device restrictions (only xperias) and repackaged the app.
He then published it on the Google play store so any device can use the app.
Problem is the package name wasn't changed so could upload as is because not all sony system apps are on Google play therefore Google play's duplicate trigger never saw the app as a problem, it didn't exist prior.
Now after this, anyone with xperias, when looking at the app, would see it as already installed on Google play. This is because even though the library on Google play does not hold some system apps, when looking at updates or install status, will look at all apps installed on device.
(This is also the case in copied games. Google play will tell you it's installed when viewed in play store)
Is it a problem:
no, the reason that your icon changed and your apps listed it as installed is because when your device looked for updates, it found the one on play but as the version numbers are the same did not update, but did update the icon to the "new" one.
Could it have been a problem:
Yes, it exposed quite a big security flaw that by publishing a matching system dumped app with new ownership, if it had enough permissions, could be "updated" to something sinister without you even knowing.
Is it sorted:
I expect so yes, I expect Google will include the names of system packages to a black list of apps that can only be updated by certain manufacturers.
Note that none of the above is confirmed by sony or Google.