The problem of redispersing suspensions and thereby homogenizing the distribution of the API has been given some attention, for instance:
Precision and uniformity of single doses are required for most dosage forms including suspensions for oral and ophthalmic administration. Therefore, solids forming sediments or aggregates must be distributed homogeneously immediately before use. If settling occurs, leading pharmacopoeias require that suspensions be redispersible by shaking [highlighting mine], [...]
(The report goes on to describe the results of a study into methods to simulate "human shaking behavior.")
The likelihood of over- or underdosing might be worth discussing with your vet (since this site or its authors are not liable for harm ensuing from measures taken as a result of content read herein), but a vigorous shake for a couple of minutes might suffice to redisperse the product into a reasonably homogeneous dispersion. There is not generally any harm in shaking a liquid formulation that doesn't specifically stipulate that it should be shaken, unless you exceed the indicated handling temperature range. This range should be stipulated in the instructions on the label. The change in temperature with shaking is a function of the ambient and package's temperatures, of container properties such as the size and thickness of the package, of shaking time, and of your dexterity. Unless you are Superman, another superhero, villain, or Santa Claus, the forces you exert on the package during shaking will not significantly alter its temperature. Shaking is therefore expected to increase the dispersity of the product and not do much otherwise. The amount of shaking required depends on the viscosity and volume of the preparation. Shake away accordingly.
Why does the stability of different formulations differ? Not all APIs are sufficiently soluble or stable in water and are therefore formulated as particles containing the API suspended in liquid. Liquid formulations are usually less stable than solid ones but a liquid formulation may nevertheless be neccessary to facilitate administration of the API. Similar issues are encountered in pediatrics (example).