From the documentation of the Natgateway
When it connects to the internet below things get used. So it makes sense to have elastic ip.

Both private and public NAT gateways map the source private IPv4 address of the instances to the private IPv4 address of the NAT gateway, but in the case of a public NAT gateway, the internet gateway then maps the private IPv4 address of the public NAT gateway to the Elastic IP address associated with the NAT gateway.