The expression "your friendly neighborhood X" (where X is a grocer, an appliance dealer, a bank, a widget salesman, etc.) has been around in U.S. English as a set phrase for at least 80 years. Here are three early examples that an Elephind newspaper database search turns up. From an advertisement in the San Bernardino [California] Sun (October 21, 1933):
The Little Store Full of Smart Suggestions Right In The Center of Everything
Why fight the down town traffic for a place to park when you can get just what you want at your friendly neighborhood store, with no time lost.
From an advertisement in the [Des Moines, Iowa] Wallace's Farmer and Iowa Homestead (October 2, 1943):
Now Be Ready With Prompt First Aid for Sudden Winter Ills!
With more and more physicians entering our Nation's armed forces, it's a wise family that is always prepared to cope with winter colds and minor ills. You can do just that by stoxking your medicine cabinet with safe, dependable McConnon home remedies and vitamin products. Like other high-quality McConnon products, these medicine cabinet necessities are DELIVERED RIGHT TO YOUR DOOR by your friendly neighborhood McConnon dealer . Get all you need when he calls!
From an advertisement in the Mexia [Texas] Weekly Herald (Decembber 8, 1944):
Drink Coffee Freely!
THERE'S NOT A HEADACHE IN A CARLOAD Just be sure of Your Coffee by Using a Brand That Has Stood the Test of Years and Is Today PLEASING CONSUMERS SAME AS ALWAYS, BECAUSE, IN SPITE OF WARTIME CONDITIONS, There Has Been No Attempt to Reduce the Quality of COOPER'S BEST COFFEE Your friendly neighborhood grocer always has a fresh supply ... He will appreciate your judgment by choosing this quality product.
Clearly, "your friendly neighborhood X" had become an advertising catch phrase long before Stan Lee brought Spider-Man to life in 1962. One of the earliest noncommercial writers to pick up on it and spread it further into popular culture was the famous advice columnist Abigail Van Buren. From "Dear Abby," in the [Palm Springs, California] Desert Sun (June 1958):
CONFIDENTIAL TO "MY DREAMS ARE GETTING BETTER ALL THE TIME:" I am not qualified to analyze dreams. If you want to know what they signify, see your friendly neighborhood psychiatrist.
In the context of the set phrase "your friendly neighborhood X," replacing the X with "Spider-Man" and incorporating the expression as a recurring line of dialogue for Peter Parker's alter ego to repeat was just a knowing bit of pop-culture humor by Stan Lee.
With regard to the humor of the expression "your friendly neighborhood Spider-Man," it should be evident from the examples given above that "your friendly neighborhood X" traditionally involves a person that every neighborhood has: an insurance agent, a car mechanic, a hardware store proprietor. But Spider-Man is unique, meaning that only one neighborhood in the world has a neighborhood Spider-Man, friendly or otherwise; and this circumstance flips the "friendly neighborhood X" formulation on its head. There's comic-book humor for you.