It really seems obvious, considering the handshake, that the hand must have symbolic meaning.
In relation to the oral ego ideal that concerns otherness itself (the representation of the first whole, instead of part object) shaking hands shows an acknowledgment of the other. Waiving hello and goodbye also have a similar form.
In coitus the vagina replaces the hand and like the hand it grasps the penis; this is reflected in the fantasies and problems characteristic of female sexuality… (Chasseguet Smirgel, Feminine Guilt and the Oedipus Complex, p. 101)
The blessing is done in such a way that with the hand that symbolizes the paternal penis the Kohenites put upon the people the divine name that equally symbolizes the paternal penis. The god-father and the people with the penis thus joined together, as it were, establish a friendship, I may say, conclude a blood-covenant.
The blessing is given by the divine hand that, as a protecting and defending phallus, rises over the people. With the putting on of the hand-phallus, that is, the divine name, of god-father upon the people, the latter's penis becomes identical with that of the father.
Feldman, S. (1941). The Blessing of the Kohenites, p. 321
He then vividly fantasied sucking the breast, made movements, approximating his thumb and index finger for which he could not at first find words, and then exclaimed: "I am biting with my finger! First, I want to bite with my mouth, then I use my fingers, and then strike with my hand!" Hand, therefore, equals a sadistic mouth as well as a sadistic penis for the patient, paralleling our previous observation that penis represented a sucking mouth in his passive fantasies. Thus we have in these fantasies of infancy the exact analogue of the patient's adult coitus: the penis (or hand) serves as a sucking mouth in the act, and as a sadistic mouth in fantasy. P. 80
Our search for an adequate explanation has led us to one major conclusion: the inhibitions are largely defenses against unconscious fantasies which represent the penis (and hand) as instruments of destructive, largely oral sadistic, drives. P. 86
Hendrick, I. (1933). Pregenital Anxiety in a Passive Feminine Character
With the boy., as pointed out, complete substitution for the micking act is gained, namely, penis equals breast, hollow of the hand equals mouth, ejaculation of semen equals milk (Rank, Genesis of Genitality, p. 134)
On the other hand in the masculine process of displacement, the penis aggressively penetrating into a hole (hand, vagina) takes on the significance of a tooth (see the masturbatory significance of tooth dreams in men which are unintelligible without differentiating between the first and second teeth). So this symbolism is a (phylo—genetic) sediment ontogenetically used as a conductor for the displacement, the driving force of which we recognize in this next stage of teething as oral sadism, manifesting itself in biting (Rank, Genesis of Genitality, 135)
To the question of why the woman's hand played so exaggerated a part I would make the following conjectural reply. Originally the hand and the mouth were the sole executive organs of the patient's aggression. (In the reversal due to the sense of guilt the hand became an organ of punishment.) At the same time the hand was a symbol for the mouth. It is true that the yielding of the semen into the woman's hand was a caricature of giving, for the fluid ran from it as from a sieve. Possibly the patient took so readily to the symbolic meaning of the hand because it contained no teeth, which would have reminded him of his own aggression. Moreover, the preference for the female hand amounted to an indirect recapitulation of infantile aggression, in identification with the phallic mother: he had wanted to strike her with his hand and now the woman was committing an 'aggression' upon him, which was in agreement with the idea of ejaculation as a punishment. At times the patient ironically lamented that it was not possible to construct a hand inside the vagina. With reference to the improbable symbolism of the hand for the mouth, see Case II (writer's cramp) in 'Der Mammakomplex des Mannes' (p.212)
Bergler, E. (1937). Further Observations on the Clinical Picture of 'Psychogenic Oral Aspermia'