How to do continuous signal processing (i.e without windowing)?
Yes, it's possible to analyse sound the way ears do.

For example, you could compute the DFT of a signal continuously using several Goertzel filters.

$$
y_k[n] = e^{j2\pi k/N} y_k[n-1] + x[n]
$$

where $k= 0,1,\ldots, N-1$, so that $y_0$ is the DC or zero frequency term.

Of course, this is an unstable filter, so some resetting or forgetting factor is needed to keep it stable. There are ways to make it better behaved numerically (see pp84-93 of Stoer &amp; Bulirsch).

The reason why we don't generally do things this way is that we are usually trying to do them in the most computationally efficient way possible.  And nature doesn't necessarily do this, so we find ways to do similar things but that work better on a computer.