What does "obtaining IP address" mean? When I try to connect to a free WiFi, my Xperia Android smartphone displays:


  obtaining ip address



Why does it happen? 
Which IP address is that? 
Is it static or dynamic? 
Does this IP address identify my smartphone?


Or does it identify the WiFi router and is its IP address common for different smartphones/laptops?
Why does it happen: When you connect to a DHCP enabled router, your device (notebook, smartphone etc) obtains an IP address in order to access internet via that router/server.

Which IP is that, static or dynamic: It's a dynamic internal IP address and is valid only within that network.

Does this IP address identify my smartphone: Yes, but only in that network and it's actually the MAC address of your network adapter that identifies your device, not your IP. 

The same is true for the router. It may have a static or dynamic IP which is given by the ISP. If it's static then yes it identifies it, otherwise no. Your external IP (router's IP) is different from your internal IP. All of the devices connected to that router have the same external IP address.