Each one is grammatically correct, but it is less common for someone to use "it."
When you use "that's," there is an implication that the phone isn't closeby (not close enough that you could point to it). Therefore the most common and natural versions are
John got a new phone. That's the second one he's bought this year.
Assuming John is not in the room looking at the phone right now. Also
I got a new phone last month. This is the third one I've bought this year.
Assuming you have the phone in your hand. Alternatively
I got a new phone last month. That is the third one I've bought this year.
If you don't have the phone with you and no one can see it.
There are other contexts in which you might use "it." For example
This is my new phone. It has a great camera.
Notice that in the US, native speakers do not normally use "mobile" as a synonym for "phone" in conversation. You might say "cell phone" if you want to distinguish from a land-line, but these days that is generally an unnecessary clarification.
You might use "mobile" to refer to someone's cell phone number if you were asking for it in a business context, like if you are a receptionist giving someone instructions on how to fill out a form. Most of the time that's just because the form says "mobile" in the space where they want your phone number.