I have several times heard scholars refer to asset pricing models (such as the CAPM) as a type of equilibrium model. Why exactly is this the case? Does this simply mean that equilibrium is a necessary condition we need to accept for the model relationship to hold? Can this be shown formally?
An equilibrium asset pricing model is one in which the asset prices jointly satisfy the requirement that the quantities of each asset supplied and the quantities demanded must be equal at that price. It is as opposed to a partial equilibrium model where the price of the asset (or at least some assets) are determined outside of the model.
In general, you can distinguish theory for a given market with off-equilibrium and on-equilibrium theory.
Off-Equilibrium theory basically means that the market is not cleared using prices, supply does not equal demand. This type of theory needs to give a good story why this is the case.
One very typical example is old Keynesian theory: due to stickiness of both wages and prices, in particular the labor market is not cleared: supply dominates demand.
The story for non clearing wages is a mixture of long-term wage contracts, efficiency wages, observed downward rigidity of wages and similar.
Asset markets however are typically assumed to be in equilibrium. The reason being that there exists a huge machinery (the whole financial system) that immediately (and mostly automatically) buys and sells assets when it predicts future changes in supply or demand.
We just don't think that asset prices are fixed at off-equilibrium values. What could possibly cause them to?