Atmega328/328P/328PA is the almost the same. At least the differences between it won't affect your arduino code.
The Atmega328PB is a different thing, avoid it if you don't definitely know what to do with it.
I don't know much about the genuine boards. Where I live, clones are the most common. Nano clones can have any controller - 328, 328P, 328PA and even atmega168/168P/168PA and 328PB
Oh okay.
So now if I get a genuine product, I expect it to have ATMEGA328.
I am just confused and want to check if there is only one Nano Part SKU: A000005 available or there are other genuine parts too available ?
Take the note that nowadays there are many different boards available under "Nano" name - classic Nano, Nano Every, Nano IoT 33, Nano ESP32...
The one that has an Atmega328P controller usually referenced as "classic Nano".
No, the genuine Arduino Nano (SKU A000005) has a Atmega328P, at least for current production (revision 3.3). There are no other active versions of the genuine Arduino Nano.
By "genuine" I mean a product manufactured and distributed by Arduino S.r.l.
Note that older revisions of the genuine Arduino Nano did use different chips.
What you get from third party manufacturers is unknown.
I still don't understand why it would matter to you. When you select Nano from the board manager it gives you a choice of either the 328P or the old 168