Will any solution to the wave equation be a wave in reality?

In the mathematical sense, a wave is any function that moves.

Not so, unless one defines the variables of the function, the above has no meaning. Wave equations are used in order to model observations of nature.

What is a wave? From sound and water waves we come to an association with sine and cosine variational behavior. Wave equations are differential equations whose elementary solutions are sinusoidal .

In classical mechanics and electrodynamics, waves are seen as the sinusoidal solutions, and involve the energy carried by the wave in space as a function of time.

When dimensions become very small, compatible with h, the Planck constant the individual "particles" electrons etc., can be described sometimes like classical billiard balls, and at the same time they exhibit a randomness, which when accumulated displays interference and other wave characteristics, which give rise to the statement "particle/wave duality".

See this answer of mine for more details.

So no, not all solutions of wave equations correspond to physical reality. One chooses the variables to be used in the wave equations so that they correspond with observables.