Yikes.
IP addresses are handed out depending on your ISP. Some ISPs (like mine) give you a static IP for fixed line connections, which means your IP will not change. I've had my current IP for several years now.
That's a completely different matter if you're at home on a router. It depends on it's configuration, but those IPs are internal to your home network, and typically begin with 192./10. Those will vary if you haven't set them to be so.
So, switching ISPs will, definitely, change your IP address at the gateway point (that being your modem). Whatever happens inside your home network, as long as the configuration is the same, will be indifferent to the ISP (in reality, it's slightly more complicated the difference is immaterial to this thread).
Now, if you use a tethering app, or Personal Hotspot, that makes your tethering device the modem. So that constitutes a change in ISP, giving your gateway device (the modem) a new IP address. And seeing as your modem would have a different configuration, you will have a different 'local' IP address.
Finally, your gateway IP address is your 'world facing' IP. It's what other computers on the internet see you as. It's unique.
And another note, mobile internet IPs are generally not fixed. You fall out of reception and back in, new IP.