One point of similarity between these two texts, which works on the level of plot and theme, is a walk through a proverbial wilderness wherein the protagonist suffers, discovers truths about himself, and ultimately is humbled by his experiences.
Holden Caulfield takes a three day journey through New York City, fending for himself and suffering from an exposure to both nature and society. King Lear follows a very similar course in the play as he wanders the wilderness suffering the abuses of both storm and man.
The suffering undertaken by both Holden and Lear can be related to identity. Each character leaves the place(s) and people that offer him a sense of fixed identity. In doing so, they come to learn that identity is a fleeting concept - more fleeting than they perhaps imagined it to be. Critically, neither character intends to actually lose his identity through his travails.
Holden wants to find a way to retain his own identity. (eNotes)
Each decides, rather, to enact his personality, to reify it, and presumes that he will be as much himself in the "wilderness" as he was at "home". This proves impossible for both protagonists, however, and each must humbly rescind his rejection of "home".
For Holden, this means that Phoebe must be accepted as his companion (not just his comfort) and he must allow himself to be admitted into an institution to be rehabilitated.
Lear's capitulation and recovery are similarly humble and humbling.
What transforms Lear after his experience in raw nature is his acceptance of his own natural limitations, of his place within, and not above, the natural world. In his reconciliation scene with Cordelia, Lear accepts that he is "a very foolish fond old man" and realizes that he is "not of perfect mind."
Having attempted to place themselves above others and beyond the need for others, Holden and Lear find that they suffer in the proverbial wilderness. Instead of proving their potency, they prove the opposite. They demonstrate to themselves the fact that society is necessary for their comfort and for the continuity of their identities.