Ah, the history of popular music does have its share of mysteries - and it seems plausible that Jack Shea never existed.
Irving (of the singing brothers Phillip, Jack, and Irving Kaufman) frequently recorded under aliases, with the agreement of his contracting record companies. Brian Rust, who listed only a handful of records he deemed of interest to jazz enthusiasts, included the aliases of Billy Clark, Sammy Burton, Harry Topping, Tom Nevill, Arthur Holt, Charles Dickson, Noel Taylor, and Brian Watt.
Kaufman was a prolific singer and performer, who made his first record in 1914 and his last record in 1974, when he was 84 years old. A good brief biography can be found on Tim Gracyk's Phonographs.
I did read a list of pseudonyms that claimed Jack Kaufman (Irving's brother) was Jack Shea. But others feel that Shea's intonation is more reminiscent of Irving's voice. Is the jury still out on the true identity of Jack Shea - or is Irving Kaufman's the voice we hear on that 1922 recording?