A device that connects to a web server always has its own IP address; the same as a PC with web browser that connects to e.g. forum.arduino.cc.
Now the question is how many simultaneous connections an esp8266 can handle. And I'm not sure about the answer. Looking around on the web indicates a maximum of 5 sockets and a compile time setting seems to limit it to 4.
Your topic has been moved to a more suitable location on the forum. Installation and Troubleshooting is not for problems with (nor for advise on) your project 
See About the IDE 1.x category.