I came upon a website that gives a very nice description and a short history of the restaurant. Here's what the website, created by Bill Bence, has to say about it:
The Lobster Palaces
After a movie at one of the downtown palaces Mimi Sheraton's affluent Brooklyn family sometimes went to The Lobster, Oyster and Chophouse, better known simply as The Lobster by its patrons, Sheraton's college boyfriend also took her there on dates in the mid-40s. It was located on West 45th Street near Times Square and had opened in 1919. Lobster houses had been a Times Square fixture for decades. Around the turn of the century they were posh hangouts, along with oyster bars, for the sporting crowd. The Lobster and its 1946 counterparts were more mid-market. Sheraton always ordered the lobster but she writes that her mother would order “strange” things like gray sole, broiled bluefish, steamed codfish or finnan haddie with an egg or cream sauceIn his 1930s guidebook Dining in New York, Rian James described The Lobster as “a low-ceilinged, rambling restaurant with the grace and courtliness of a one-arm cafeteria; with rushing, ribald waiters, who dash up and down between the long aisles of tables with squirming lobsters in their hands, who take your order in a restless, 'must be getting away' fashion, making the distance between the oyster bar, up front, and the kitchen in the rear, in pretty nearly nothing flat.” The walls were decorated with mounted lobsters and fish and cartoons from Harry Hershfield and Fay King. According to Rian, it also had the best seafood in the city at a reasonable prices, which drew mobs of suburban and outer borough theatergoers in such numbers that people waited on line on the sidewalk to get in. The many other lobster houses in the immediate vicinity based their business on the overflow.
The Lobster was among a number of establishments that were fined in February 1946 for charging customers more than the legal ceiling prices set by the OPA. The Lobster paid a much higher fine than the other restaurants cited. That summer it also was cited for unsanitary conditions. It stayed in business until 1972 when increased costs, declining patronage and a change in the neighborhood made it no longer profitable. It was a favorite lunch spot for the staff of The New Yorker and Richard Harris wrote a "Reporter At Large" piece in the December 30 issue about its closing. To him it was a "comfortably unattractive," bustling place with efficient waiters and the air of convivial private club where you could get simply prepared, fresh seafood at reasonable prices. The owners, who were really pissed at the unions as well as the city bureaucracy, told Harris that the unions used to block the employment of African-Americans from any but menial positions. They defied the unions to promote a Black employee to the oyster bar.
The back of the postcard seems a little odd, since it has a return address stamp from New York, but was postmarked in Astoria, Oregon...and there's no message. It's not quite as strange though if you know that the recipient, Edwin Payne, was a postcard, stamp, and cover collector and a postal historian. Here's a plaque in his honor from The Salem Stamp Society.