Too long for a comment, but, I am afraid, not very exciting . In the subject-predicate logic (syllogistic) both subjects and predicates were of a kind, assimilated to classes (so "all men are mortal" included the class of men into the class of mortals). Since Frege needed multi-place predicates, assimilation to classes did not work for him. So he drew a line in the sand between objects (Man) and concepts (is mortal). Why not subjects and concepts, you ask? Because in the meantime Kant inverted the traditional usage of subject and object, and now we have a messy mixture of the two usages. But the point is that we can gloss Johnny loves Sally in terms of a single concept (Loves) and two objects (Johnny and Sally), convert them into a two-place predicate and two subjects, and a path to polyadic quantificational logic is open. It should be noted that Frege's "objects" are close to Kant's "intuitions" in a peculiar sense of "individual ideas", as in Logik (1897), not necessarily tied to sensibility, as in the Transcendental Aesthetic of CPR. Wikipedia has an article on Frege's Über Begriff und Gegenstand (English translation On Concept and Object).
Information Philosopher gives English translations of passages from Über Begriff und Gegenstand. Here is the one concerning Gestalt, translated as "shape":
"If the sign "a" is distinguished from the sign "b" only as object (here, by means of its shape), not as sign (i.e., not by the manner in which it designates something), the cognitive value of a=a becomes essentially equal to that of a=b, provided a=b is true."
It does not seem that Frege's use is technical, Gestalt simply refers to a property of sign's token, its shape. "Gestalt" has been used non-technically (outside of Gestaltpsychologie) at least since Kant, and since it is Kant many philosophers feel obligated to fit the word somewhere. Kant uses it loosely in the notorious transcendental schematism section of CPR, "which is famous for its profound darkness, because nobody has yet been able to make sense of it" (Schopenhauer):
"Der Begriff vom Hunde bedeutet eine Regel, nadi welcher meine Einbildungskraft die Gestalt eines vierfüßigen Tieres allgemein verzeichnen kann, ohne auf irgend eine einzige
besondere Gestalt, die mir die Erfahrung darbietet, oder auch ein jedes möglidie Bild, was idi in concreto darstellen kann, eingeschränkt zu sein". (A141/B180)
I altered the Guyer's translation below to bring out the use of Gestalt (which he also translates as "shape"):
"The concept of a dog signifies a rule in accordance with which my imagination can specify the Gestalt of a four-footed animal in general, without being restricted to any single particular Gestalt that experience offers me or any possible image that I can exhibit in concreto."
Kant later terms this Gestalt by the better known term of concept's "schema". Schemata are something like Plato's intermediaries between forms and sensible things, but in Kant's case, within us rather than without, and between concepts and intuitions. The modern consensus is that Kant's schemata/Gestalts have something to do with perceptual pattern recognition, and hence with framing of perceptual concepts, presumably. This is where the agreement ends. If you want to plunge into the "profound darkness" of the schematism try Pendlebury's Making Sense of Kant's Schematism. I do not think it succeeds at that, but it is a valiant attempt. And here is what Kant wrote right after the above quote for encouragement:
"This schematism of our understanding with regard to appearances and their mere form is a hidden art in the depths of the human soul, whose true operations we can divine from nature and lay unveiled before our eyes only with difficulty."