Albert,
Your 3-phase power is probably 120/208. My guess is your reading 120v phase to ground, but if you put a good meter phase to phase it probably reads in the neighborhood of 208v. Check it with a high-grade meter (Fluke) sometime. This is a typical 3-phase voltage used in the USA no matter where you live.
If your reading 240v phase to phase then you probably have 120/240 with a high leg. That means one of your legs will be reading 277v phase to ground. I doubt you have this kind of power in a residential neighborhood.
In a typical residental service the panel does derive power from two separate 120v supply lines, but the supplying transformer is a single-phase transformer capable of producing 240volts. (This does not apply to you)
It does not take two transformers to produce 120/240
And without two or three transformers you cannot get two phase (No such thing) or three-phase power.
If Memory serves the transformer on the pole has a center tap (Neutral/ground), which allows for a reference point on either leg. Thus there are not two phases present just two hot's and one Neutral (Ground reference) derived from one transformer (One phase of a three phase system) on the pole
Hot to hot reading 240v
Hot to neutral reading 120v (when grounded to earth)
(Again this does not apply to you)
Your house has one main drop three phase derived from three single phase transformers on a pole or in an underground vault near by. These three transformers are configured to produce the desired voltage you seek which is typically 120/208 based on what you've told me already. This is in fact your main power at your main panel. 120/208 three phase.
From that three phase panel two hots and a neutral are sent to a single-phase panel (in the same closet). Even though we typically refer to this panel as single phase (Standard terminology) you are in fact feeding it with two phases. (ONLY BECAUSE YOU HAD 3 PHASES TO START WITH) This is a little different than a normal residential installation and in fact would be treated as a commercial installation.
Let's talk about the reason the two panels inside your house have different voltages. They are 120/240 panels which means you are using *A phase* and *B phase* tapped from your main three phase panel to feed one. *B phase* and *C phase* tapped from your main 3 phase panel to feed the other. You see that each panel has at least one different phase than the other. That's why the voltage reading is never the same at both panels.
It would be a little hard for me to believe that youre only reading 120 phase to phase at these panels. You should be reading 208 phase to phase. There is no way an electrical contractor would do this type of installation unless he doubled the size of the supply neutral, you would also have to run a dedicated neutral with every hot comming out of the panel.
As for your meter That's called a CT meter (Current transformer) when you pull the meter you do not disconnect the line power, just the ct's. Pretty common in Commercial installations.