Intuitively, why is $\langle V \lvert W\rangle$ "square" of norm $ \lvert V\rvert$ and not $\lambda \cdot \lvert V\rvert$?

Question

I wanted to gain an intuitive proof of the Schwarz Inequality and Triangle inequality (primer for Quantum Mechanics).

I intuitively interpreted inner product, $\langle V \lvert W\rangle$, as the measure of likeness, as "How much $\lvert W\rangle$ is like $\lvert V\rangle$ ?". The norm comes to play when I think of, $\langle \frac{V}{\lvert V\rvert}\lvert W\rangle$, as if "a single tulip is less alike to a field of tupil, than what a bouquet of tupil will be" (say if, $\lvert V\rangle$ is a bouquet and $\frac{\lvert V\rangle}{\lvert V\rvert}$ is a single flower).

I am comfortable with this concept of scalar measure as long as comparing distinct objects or generalised vectors. But my intuition breaks down as I think "how much a bouquet of rose is alike to a bouquet of rose"; and the intuitive answer seems to be $\lvert rose bouquet\rvert$ and not its square.

I'm not sure if pursuing such intuitive reasoning will do much good with my course ahead,
but still, what am I missing here?