‘Pet bank” is a derisive term that was applied to state banks that were selected to receive government deposits in 1833, taking the place of the Second United States Bank that President Andrew Jackson strenuously opposed. The term “pet” implies “favorite” and is similar to “crony.” Although the “pet bank” term is historical, it could be applied to any bank that receives funds through cronyism and special favors.
   
“Pet bank” has been cited in print since at least early November 1833.
     
 
Wikipedia: Pet banks
Pet banks is a pejorative term for state banks selected by the U.S. Department of Treasury to receive surplus government funds in 1833. They were also named “Wildcat Banks”. They were made among the big U.S. bank when President Andrew Jackson vetoed the recharter for the Second Bank of the United States, proposed by Daniel Webster and Henry Clay four years before the recharter was due. Clay intended to use the rechartering of the bank as a topic in the upcoming election of 1832. The charter for the Second Bank of the United States, which was headed by Nicholas Biddle, was for a period of twenty five years beginning January 1816, but Jackson’s distrust of the national banking system (which he claimed to be unconstitutional) led to Biddle’s proposal to recharter early, and the beginning of the Bank War. Jackson cited four reasons for vetoing the recharter, each degrading the Second Bank of the United States in claims of it holding an exorbitant amount of power.
 
The term implied that the state banks were controlled by Jackson. By 1833 there were 23 “pet banks” or state banks with US Treasury funds. The term gained currency because most of the banks were chosen not because of monetary fitness but on the basis of the spoils system, which rewarded political allies of Andrew Jackson.
 
Merriam-Webster Dictionary
pet bank noun
Definition of PET BANK
: any of a group of state banks selected as depositories of federal funds removed from the U.S. Bank during the first Jacksonian administration
 
9 November 1833, Daily National Intelligencer (Washington, DC), pg. 3, col. 1:
On the other hand, let us inquire what course has been pursued by the pet banks. (...) When the time comes for drawing on , we shall see how they will bear it, and how the Secretary of the Treasury will get along with them.
 
13 November 1833, The Globe (Washington, DC), pg. 2, col. 2:
“When the time comes for drawing on the pet banks, we shall see how they will bear it, and how the Secretary of the Treasury will get along with them.”
 
18 November 1833, New-York (NY) American, pg. 2, col. 2:
But on Saturday, the Bank of America—a pet Bank and an honest!—having the Branch in debt $100,000, immediately sent $80,000 of the amount in specie.
 
Google Books
23 November 1833, Niles’ Register, pg. 194, col. 2:
There will be many brokers-shops established, and perhaps, by some of the “pet banks” themselves! in those distant neighborhoods in which their notes are forced on the people.
 
Chronicling America
3 May 1836, The State Journal (Montpelier, VT), pg. 2, col. 3:
No one pretends that the thirty-five millions of dollars, now in the vaults of 35 pet banks, is needed for any governmental purposes.
 
Google Books
A Dictionary of Americanisms on Historical Principles
Edited by Mitford M. Mathews
Chicago, IL: University of Chicago Press
1951
Pg. 1227:
pet bank, (see quot. 1914). Obs.
1834 C. A. DAVIS Lett. J. Downing 353 Mr. Van Buren…didn’t say nothin about hard money in the place [of bank bills]...The safety fund banks and the pet banks couldn’t drink that toast no how.
1852 GOUGE

p. vii. Does the deep experience of the evils we have suffered under both a national bank and a league of “pet banks” incline us to separate bank and State?
1914 Cyclo. Amer. Govt. II. 674/2 Pet Banks. Term derisively applied by the opponents of President Jackson to the state banks which Amos Kendall and Secretary Taney selected in 1833 in which to place the government deposits, in place of the Second United States Bank.