Do I need to buy an ISBN to publish on KDP?
Updated
No — KDP assigns a free ISBN to paperbacks (ebooks don't need one on Amazon at all). Buying your own ISBN matters only if you want to be listed as the publisher of record or plan to distribute the same edition beyond Amazon.
The free KDP ISBN lists 'Independently published' as the publisher and can only be used on Amazon. For most first books that is a perfectly fine trade.
Buy your own ISBN (from Bowker in the US, Nielsen in the UK) when you want your imprint name on the record, or when the same edition will be sold through other channels — each channel edition needs consistent identification.
Either way the ISBN is metadata, not formatting: your interior PDF and EPUB do not change. Add the number to your copyright page if you have one at typesetting time.