The phrase to rain cats and dogs and to hail cabs (or taxis, etc.) is a jocular extension of the phrase to rain cats and dogs, meaning to rain very hard.
This jocular phrase occurs, for example, in Kid News, published in The Southern (Carbondale, Illinois, USA) of Monday 12th April 2021:
What is worse than raining cats and dogs?
Probably coined on various occasions by different persons, independently from one another, the phrase to rain cats and dogs and to hail cabs (or taxis, etc.) puns on:
– the verb hail, meaning to pour down like hail, i.e., like small pellets of ice;
– the verb hail, meaning to call out (a cab).
The earliest occurrence of the phrase to rain cats and dogs and to hail cabs (or taxis, etc.) that I have found is from Punch, or the London Charivari (London, England) of Saturday 12th April 1856—here, the phrase is part of an extended metaphor based on the image of a flood (i.e., a great influx) of persons:
A Rare Shower.
A Gentleman, who had been in the City at four o’clock, when a flood of clerks is generally let loose, was talking upon the subject to his wife at dinner, when he wound up grandly by saying, “I never witnessed such a scene of confusion in all my life, my dear. What with the clerks, and what with the rain, I never shall forget it! Only imagine, my darling, that not only was it raining cats and dogs, but hailing omnibuses also at the same time.”
The second-earliest occurrence of the phrase to rain cats and dogs and to hail cabs (or taxis, etc.) that I have found is from the humoristic column Atlas With His Load Off, published in The Atlas. A General Newspaper and Journal of Literature (London, England) of Saturday 30th January 1858:
DEGREES OF COMPARISON IN CONUNDRUMS.
Bad. What extraordinary animal production may be procured in the Isle of Wight?
Mutton from Cowes 1.
Worse. What is worse than raining cats and dogs?
Haling [sic] cabs and omnibuses!!
Worst. What is the difference between a cat and a document?
One has clawses at the end of its pawses, and the other has pauses at the end of its clauses!!!
1 Cowes is a seaport on the Isle of Wight.
Later in the 19th century, the phrase to hail omnibuses was appended to the phrase to rain pitchforks.
The earliest occurrence of this variant that I have found is from Varieties, published in The Dundalk Democrat and People’s Journal (Dundalk, Louth, Ireland) of Saturday 4th June 1864:
What is equal to raining pitchforks?—Hailing omnibuses.
The second-earliest occurrence of this variant that I have found is from the Portland Daily Press (Portland, Maine, USA) of Monday 21st November 1864:
An atrocious conundrum from the Boatswain’s Whistle 2: What is worse than raining pitchforks? Hailing omnibuses.
2 The Boatswain’s Whistle was a short-lived newspaper, published exclusively for the National Sailors’ Fair held in Boston, Massachusetts, in November 1864.
The Boatswain’s Whistle’s “atrocious conundrum” was jocularly expanded in the following from Wit and Humour, published in the Stockton Herald, South Durham and Cleveland Advertiser (Stockton-on-Tees, Durham, England) of Friday 24th February 1865:
The ‘Boatswain’s Whistle’ proposes the following:—What is worse than raining pitchforks?—Hailing omnibuses. We prefer reigning belles.