You are right that the distinction between the politician’s art and the kingly art is obscure. The immediate context of the dialogue does provide a partial answer but it’s important to understand the dialogue as a whole and in the context of other dialogues in order to understand this remark.
We learn from Socrates about Cleinias’ proposal that generalship is the supreme art. The problem is that:- “… When they (sc. generals) have hunted either a city or an army, they hand it over to the politicians—since they themselves do not know how to use what they have hunted…” (290d)
The idea that political and kingly art are the same is a hypothesis, arrived at while Socrates and Cleinias were considering whether the “kingly art” is “what provides and produces happiness”; they supposed that the statesman's and the monarch's arts were one and the same and, comparing medicine, which produces health, and agriculture, which produces food, they ask what the political/kingly art produces?.
They had no answer. Socrates continues:- “So then I myself, Crito, finding I had fallen into this perplexity (aporia), began to exclaim at the top of my voice, beseeching the two strangers as though I were calling upon the Heavenly Twins to save us, the lad and myself, from the mighty wave of the argument, … ” (Euthydemus, 293a)
There’s no return to the relationship between politics and kingship.
The essential point is that they never consider the difference between the two, only the hypothesis that they are the same. Since you need a king to practice the art of kingship, there is a problem in democratic or oligarchical states. It would make sense that the art of politics is simply the art of kingship in a state that has no monarch and vice versa.
So what are we to make of this?
Plato’s project in this dialogue is not to understand politics or monarchy. It is to show up Euthydemus and Dionysodorus, who are professional teachers, skilled in eristic argumentation – what we would now call sophistry. It is, essentially, a satire, but framed by the serious issue that Crito is considering whether to place his son with them; this adds to the bite. By the end of the dialogue, Socrates has to persuade him to “be reasonable, Crito, and do not mind whether the teachers of philosophy are good or bad, but think only of philosophy herself. …. ” (ibid, 303a).
Jowett’s introduction to his translation treats it as something of a historical relic, on the grounds that we now understand logic and have no problem seeing through sophistical tricks. I think there’s more to it than that.
As the dialogues around the trial and death of Socrates show, it was very important to Plato that he rescue Socrates’ reputation, as well as articulating his own philosophical project as distinct from his tradition. He needed to distinguish philosophy very clearly from rhetoric on one hand (see Gorgias) and sophistry on the other – and that’s what this dialogue is about.
Socrates experiences aporia himself, is on the receiving end of sophistry, and ends up defending himself with the weapons that Euthydemus and Dionysodorus use against him - unsuccessfully.(303a) The difference in terms of the drama is that for Euthydemus and Dionysodorus dialogue is a game and their aim is to win. But Socrates is sincerely interested in philosophy; this is not a game to him and he is not simply trying to win. Socrates’ aporia here may or may not have actually been shared by Plato as his thought developed; but either way, his readers will know that the questions raised here are not abandoned, like an old football, but followed up (Republic, Statesman, Laws).