tl;dr: It's complicated.
You've identified some valid relationships between evaporation and solubility, however, it's not so much that evaporation into air is a form of solubility, but rather both evaporation and solvation are forms of phase change.
I think this idea is best illustrated by looking at solid-state solubility: that is, dissolution of solids into other solids. Below is an image of the iron-carbon solid solution phase diagram.

If you've only done introductory Chemistry you've probably only ever seen phase diagrams of temperature plotted against pressure. We can also consider the "concentration" of different chemical elements or molecules in our phase diagrams as well. Here, we have the carbon concentration plotted on the $x$-axis as a mass concentration. On the $y$-axis we have temperature. This is a fairly busy plot, but there are a few features worth pointing out.
For low carbon concentrations, carbon can dissolve completely into the iron structure: that is, carbon is soluble into the iron solvent. With too much carbon, the material becomes a mixture of pure iron and iron carbide (Fe$_{3}$C). The diagram cuts out at 6.67%-mass carbon, which corresponds to pure Fe$_{3}$C. At higher temperatures different structures become favorable, including different phases of solid iron (that is, different arrangements of iron atoms).
The point here is that solution of one compound into another is a phase change - that's why you can see similarities between evaporation and solution. When dissolving a solid into a liquid, the solid will dissolve up to the solubility limit of the system (incl. temperature and pressure), with any excess remaining as a separate phase. This is a general phenomenon with phase changes.
Now, if you're evaporating a liquid into a vacuum the system will also have some finite "solubility limit" (if you insist on thinking of it that way). Note that this is a property of the system, not the liquid or vacuum! If you add more compounds to the system, you are effectively adding more dimensions to your phase diagram (and more complexity to boot!). In general, things like vapor pressure are relatively insensitive to the chemical composition of a gas, so we can approximate them as being independent. There are exceptions! If the vapor reacts chemically with the gas you end up with another dimension in your phase diagram where you need to account for the (potentially 2-way) reactions within the gas itself and the overall equilibrium of the system.
To try and put it as simply as possible: you can think of a phase diagram as describing the chemical equilibrium of a system, in the broadest possible sense.
This is a topic that might be covered over half a course at undergraduate level. Hopefully this answer gives you some sense of where to look for further reading.