India gives us one of the easiest examples in the world.
Amul.

Think about Amul and chances are you don’t picture only the red wordmark, you think of the Amul girl.
The topical ads. The humour. Butter. – “The Taste of India.”
A certain kind of familiarity, decades of seeing the brand show up consistently in Indian life. That is the brand!
The logo helps you recognise it, the rest gives the recognition meaning.
Amul’s long-running topical advertising and consistent mascot have become central parts of how the brand is remembered.

Paper Boat is another good example.
Its identity was not simply a nice logo applied to a pouch. The brand was deliberately constructed around nostalgia, childhood memories, traditional Indian drinks, naming, tone of voice, illustration and an unusual packaging format. Elephant Design describes the original brief as building an emotional connection around familiar Indian flavours and childhood memory.
Take away the logo and much of Paper Boat would still feel like Paper Boat.
That’s when a brand system is doing its job.
Now imagine changing only the Paper Boat logo while keeping:
the product,
packaging shape,
illustration,
language,
stories,
photography
and experience.
Would it suddenly become a completely different brand? Of course not. The logo matters. But asking the logo to carry the whole brand is like expecting somebody’s signature to tell you everything about their personality.
Your logo identifies you.
Your behaviour explains who you are.