Why is the concept of a “state covariance matrix” necessary in estimation?

I'm currently taking a course in optimal estimation (and it's still very early in the course).

Much of our work is based around the idea of a measurement model

$$y=Hx + v$$

This model assumes our measurement $$y$$ is a function of some "state" and measurement noise.

Now when attempting to estimate the state $$x$$ in a stochastic least squares sense, one of the goals is to minimize the trace of the state covariance matrix $$P$$.

My question revolves around the idea of a state covariance matrix. Why exactly does it make sense to assume such a thing exists? By doing so we making the assertion that our "state" is a random variable.

Let's say we are trying to estimate the resistance of an unknown resistor by inserting a known voltage and measuring the current. In this case, our "state" is the resistance. Why would it make sense to assume our resistance is a random variable?

Do we assume the state is a random variable since there might be uncertainty in our model?