5.7Not all tonic-family chords are I

In C major, these chords often have tonic-like behavior:

I = C

iii = Em

vi = Am

Why?

Because they share notes with C major.

C:

C E G

Em:

E G B

Am:

A C E

Both Em and Am share two notes with C major.

So they can sometimes substitute for tonic function without sounding completely unrelated.