Solution 1:

According to Jack Lynch, whose book The English Language: A User's Guide is well worth the modest investment for those without the patience to deal with the OED or Fowler,

Many people get spooked by the plurals of proper names, but the rules aren't that different...The only difference between proper and common nouns is that the proper names ending in -y shouldn't change form in the plural: just add an -s. The members of the Percy family are the Percys, not the Percies.

Lynch goes on to warn against pluralizing with punctuation:

Resist the urge to put an apostrophe before the s in a plural, whether in common or proper nouns. The term for this vulgar error is the "greengrocer's apostrophe," from the shopkeepers' habit of advertising their "potato's" and "apple's."

If one has in one's company a multiplicity of persons named Jess, then indeed they are "Jesses". The same spelling would apply in the case of two or more persons yclept "Jesse." They too would be Jesses.

The questioner asserts that "English is silly," which I think is an unkind thing to say about anyone's Mother Tongue. English is sometimes fickle, sometimes illogical, occasionally intractable, but it is capacious, flexible and alive. It tolerates confusion. In figurative and humorous usage, English even thrives on confusion. Comfortable speakers of the language use it as a platform for invention and improvisation.

In the questioner's example, wherein we have a hyperabundance of Jesses of unspoken origin or character, it is likely that one or more of them will have adopted or been given a nickname by their peers: "Spike," "Peach," "Two-Step" or "Four-Eyes." The possibilities of human interaction, mediated by English, are nearly endless!

Solution 2:

The question in your title is already addressed elsewhere, e.g. the site Rathony mentioned in comments (here and there). This answer addresses the ambiguity between the two sets of Jesses.

Words that look the same on the printed page but differ in meaning are called homographs. The inherent ambiguity is part of its nature - they won't be homographs otherwise. They appear in other languages as well: the linked page shows some Chinese examples, and here's another page that purports to show some French examples (though many of them look like English words to me).

Sometimes, very little context is required to disambiguate them. For example, St James St is clearly Saint James Street without the need for further information.

There are occasions where ambiguity is acceptable. You cite the example, "Both Jesses are dating both Jesses." Since both instances of both are not elaborated upon, I'll consider the identity of each reference to "both Jesses" to be prior knowledge. Although we don't know which group of "both Jesses" a particular Jess/Jesse is part of, it doesn't matter here because if one dates the other, then the latter automatically dates the former.

There will also be times when more context is required, such as with your example involving cats and dogs. That's simply the nature of homographs and language. In your dating example, you don't object to the ambiguity of which specific Jess/Jesse dates whom, a matter what is (conceptually, at least) distinct from the issue of homographs - it would still be ambiguous if the Thomases dated the Joneses.

With a rich language like English, ambiguity can sometimes be part of the fun. As the Thomas/Jones example shows, you can have ambiguity even without homographs. The richness of the language also allows other forms of expression when precision is needed and ambiguity is a liability.