Android - On-device calendar exists?

The native AOSP calendar app was replaced by Google's own1. It went by the package name, but is no longer found in either Google Play Store or on F-Droid (you can find its code on Github, though). As for synchronization, that's usually not done by the calendar app itself but by the corresponding sync service; most calendar apps use the "calendar provider" of the Android system to store their data.

If you want a "local calendar" that's not synchronized at any cost, and your device doesn't offer such, you can e.g. use Offline Calendar; but that app doesn't seem to be maintained anymore (last version is from 2013).

1: See Google’s iron grip on Android: Controlling open source by any means necessary

Google Calendar could be considered as native android calendar app. You can download it from Google Play. But I personally don't like it.

As for device-only calendar - I have never seen an android phone without such a calendar (I'm not familiar with Samsung devices though). Normally you can choose a calendar during creating an event.

You could try Offline Calendar. App description says

Lets you add calendars to the Calendar app, which are not synchronized and are accessible only on the device itself. Create the calendar in this app and it will then be available in the Calendar app.

I had tried it earlier, but gave up since I found interface not very friendly (for me at least-your experience may be different). Exporting to new device could be as you would do for other apps (backing up and reinstalling) or as I see in developer reply suggesting interface with another app to export(see 15/09/2015)

EDIT: I posted this as a comment and by the time typed the reply, found another answer (by Izzy) recommending this !