When NAT router is used for global address translation, two networks may have same private IP address. Am I right or wrong?
My internal private IP addressing and another person's internal IP addressing may be the same or similar, yes.
On a NAT router or Firewall, you have an inside interface and an outside interface. Using an ACL, or depending on your manufacturer's OS, you determine what internal traffic source is NAT'd to an external address when it crosses the Firewall or NAT router.
Out on the internet your public addresses are bought and assigned; there are no overlaps in Public address space. Overlaps in Public address space would cause serious problems.
Yes, you are correct. This is why it is critically important in business to consider this in network architecture planning. It's awful after a merger to create VPN links between offices when they all use 10.0.0.0/8 for instance.
Keep your prefixes as small as you require (with space for predicted growth), and if you are setting up multiple offices, even if you think they'll never be connected, use different prefixes. Also, use an IPAM software or at minimum a spreadsheet to keep track of what prefixes are where.
You don't want to have to do multi-layer NAT and other trickery to connect overlapping subnets, unless absolutely necessary. Guaranteed you'll get "we can't change our IPs", so what you're left with is a hard-to-document setup that resembles spaghetti.