In this case the mobile acts as a router??
It actually does.
Who assigns my laptop it's private ip address?
The phone does – it runs a DHCP server, just like a "full" home router would, and the laptop gets the address through a DHCP lease.
Is the private ip address of laptop the same as private ip address of my mobile?
No. They're within the same subnet, though, but they're separate devices and both have their own unique addresses in that subnet.
Can i change the private ip address of my laptop and be connected to the same wifi??
As long as the new address is still within the same subnet, and doesn't accidentally conflict with the router (i.e. the phone) or some other connected device – yes.
Does my laptop has public ip address?
No. It'll appear to have one when accessing Internet sites, but the translation (NAT) is done within the router (i.e. the phone) – the laptop is actually unaware of it.
If private IP addresses are same within the wifi network how does it identify different devices?
They're not the same.
If i go to settings in mobile it shows IP address as 100.96.79.125 is it private address or public ip address ?
It's a private address (from the special 100.64.0.0/10 range that's specifically meant for assignment by ISPs rather than home networks).
Most mobile operators only assign this type of address – and even many fixed-connection ISPs are no longer assigning public addresses.
Will it have public ip address
In your case no, same as above. The routers at your ISP will also perform NAT and automatically translate the phone's address to a public one, allowing connections to Internet, but that public address doesn't actually belong to the phone itself. It may even be shared between several customers.
(A phone could have a public IP address in theory, but I don't think there are any mobile operators which assign one anymore. Our Tele2 actually used to do this, but I think even they switched to CGNAT and private addresses last year...)