This might help for your intuition.

The assumption that you mentioned is only used for deriving the capacitance of a capacitor(using Gauss law). But in reality capacitors always have equal and opposite charges on it's plates while connected in a circuit (watch the video).

Why does the field confined only inside the capacitor?

It's indeed an application of Gauss law! I would like to give a hint, so that you could build other things on your own. Consider two infinitely large charged sheets(oppositely charged) kept at some distance parallel to each other. Now, what is the electric field inside and outside the charged plates?

Remember, in reality, capacitor plates are kept very close, so that a point inside the capacitor see the plates as (so called)infinitely large.

This might help for your intuition

The assumption that you mentioned is only used for deriving the capacitance of a capacitor(using Gauss law). But in reality standard capacitors always have equal and opposite charges on it's plates while connected in a circuit (watch the video).

Why does the field confined only inside the capacitor?

It's indeed an application of Gauss law! I would like to give a hint, so that you could build other things on your own. Consider two infinitely large charged sheets(oppositely charged) kept at some distance parallel to each other. Now, what is the electric field inside and outside the charged plates?

Remember, in reality, capacitor plates are kept very close, so that a point inside the capacitor see the plates as (so called)infinitely large.