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The noun bingo cage designates a device with a revolving cage or drum, used in a game of bingo to mix up the numbered balls or slips, or for drawing numbers or prize tickets in a lottery, tombola, etc.
The following picture is from an advertisement for a bingo cage set, published in the Hawaii Tribune-Herald (Hilo, Hawaii, USA) of Sunday 16th December 2007 [Longs Drugs supplement: page 5]:

—Cf. also the satirical phrase the (great) god Bingo and the offensive phrase bingo wings.
These are, in chronological order, the earliest occurrences of the noun bingo cage that I have found:
1-: From the following advertisement, published in The Courier-Journal (Louisville, Kentucky, USA) of Saturday 27th March 1937 [section 2, page 8, column 6]:
BINGO GAMES—For churches, organizations: 30 to 3,000 different cards, Bingo cages and round numbers. Low prices. K. S. Caufield, Inc., 308 S. 3rd.
2-: From the following advertisement, published in the Cleveland Plain Dealer (Cleveland, Ohio, USA) of Sunday 18th April 1937 [page 6-B, column 3]:
BINGO CARDS, BINGO CAGES and public address systems for rent. HE. 9398. Write E. Peters, 1964 E. 73.
3-: From Our Brisbane Letter, published in The Toowoomba Chronicle and Darling Downs Gazette (Toowoomba, Queensland, Australia) of Thursday 2nd September 1937 [page 9, column 3]:
Bingo Scandal to End
Every decent-minded person in Queensland will be glad that the Government’s hand has been forced in the direction of putting a ban on the wretched bingo ramp. It has been a public scandal for [so] long that while grown men can be run to court for taking part in the perfectly fair even-money gamble of two-up, kiddies of tender years have been allowed, and in many cases actively encouraged, to stake their modest sixpences at bingo, where the odds are in favour of the promoters to the extent that approximately half the money wagered in one round by the punters is lost to the “bank.” For a long time past there has been a steadily growing feeling against the game, […] and the last straw came this week when a city daily published a full page of flashlight photographs taken at several bingo games operating in the city. These photographs showed men, women, youths, young girls and children sitting around the benches, demonstrating beyond doubt the fact that it was inducing the gambling spirit in the young, and enticing people to bet who obviously could not afford to lose. One poor little kiddie was sitting, half asleep, beside a woman intent on the game. This exposure proved the death knell of the bingo racket, for to-day a general prohibition was announced, and within about a week the bounce of the wooden ball in the bingo cage will be a memory.
4-: From the following advertisement, published in the Pottstown Mercury (Pottstown, Pennsylvania, USA) of Monday 25th April 1938 [page 1, column 6]:
SWOYER’S NOVELTY COMPANY
81 Mt. Vernon St. Phone 2103-J
Fair, Carnival & Bazaar Supplies
Bingo Cages, Balls, Master Boards
5-: From Legion Bingo Raided, published in The Pittsburgh Press (Pittsburgh, Pennsylvania, USA) of Monday 6th June 1938 [page 4, column 5]:
The raid last night followed by several weeks an announcement by the District Attorney and the Public Safety Director that henceforth bingo games would not be tolerated. […]
[…]
Lieut. Pischke said about 75 women were playing when he entered the Legion Hall […].
The officer said the women were seated at tables and numbers were being called by Mrs. Gertrude Pillion, of 413 Mansion St. He confiscated a bingo cage, similar to a bird cage in which dice are shaken, and a bingo card. They were brought into the courtroom as evidence.