“First dog” is a term sometimes used for a dog of the president of the United States. “DOTUS” (Dog Of The United States) has also been used.
   
“The big dog Hector is Mrs. Cleveland’s special pet, and so long as she was at home he behaved himself as befitted the first dog in the land” was printed in the Sioux City (IA) Daily Journal on June 15, 1887. “WASHINGTON, June 16.—The “first dog in the land,” as Mrs. Cleveland’s black poodle Hector is called” was printed in The Daily Argus-Leader (Sioux City, IA) on June 16, 1887. “Standing as the undisputed head of the exclusive dog circle is Caro, a royal Italian poodle, the pet of Miss Helen Taft. This ‘first dog of the land’ is a privileged character at the White House” was printed in the Washington (DC) Post on January 12, 1913. “The nation has a new ‘first dog’—a golden retriever puppy named Liberty which flunked President Ford’s first test by showing a preference for publicity over work” was printed in the Philadelphia (PA) Inquirer on October 7, 1974. “totally saw Beau at the white house!!! #firstdog” was posted on Twitter by Jami on December 11, 2009.
   
“First Cat” (cat of the president of the United States) is a similar term.
   
A list of the pets of the United States presidents is here.
     
   
Newspapers.com
15 June 1887, Sioux City (IA) Daily Journal, pg. 1, col. 4:
The big dog Hector is Mrs. Cleveland’s special pet, and so long as she was at home he behaved himself as befitted the first dog in the land.
   
Newspapers.com
16 June 1887, The Daily Argus-Leader (Sioux City, IA), pg. 1, col. 4:
HECTOR PROVES AN ALIBI.
Mrs. Cleveland’s Pet Poodle is not a Rabbit and Chicken Slayer, But a Goo Dog.
WASHINGTON, June 16.—The “first dog in the land,” as Mrs. Cleveland’s black poodle Hector is called, has a friend and ally in the white house in the “first cat in the land,” who is pure white when she doesn’t take advantage of her mistresses’ absence to search for plebeian rats an mice in the coal hole of the executive mansion.
 
7 February 1902, The Courier-Journal (Louisville, KY), “heard and Seen in Washington,” pg. 4, col. 6: 
First Dog In The Land.
At Oyster Bay Jack was simply the yellow dog of the Roosevelt family. Now he is Jack Roosevelt, the first dog in the land.
 
12 January 1913, Washington (DC) Post, “Elite Dogs of Society Keep Gentle Hands busy,” second part, pg. 1, col. 2:
Standing as the undisputed head of the exclusive dog circle is Caro, a royal Italian poodle, the pet of Miss Helen Taft. This “first dog of the land” is a privileged character at the White House, where he can be seen daily walking or napping in the historic rooms of the executive mansion.
   
13 March 1921, New York (NY) Tribune, Harding Besieged by Would-Be Officeholders Bearing Gifts,” pt. 1, pg. 1, col. 4:
First, of course, there is Laddie Boy, the Airdale First Dog of the Land.
 
7 October 1974, Philadelphia (PA) Inquirer, “The Scene in the Nation and the World,” pg. 3-A, col. 2:     
The nation has a new “first dog”—a golden retriever puppy named Liberty which flunked President Ford’s first test by showing a preference for publicity over work.
 
Google Groups: alt.fan.dan-quayle
Dave Barry & Dan Quayle (again)
Maddi Hausmann
Dec 30, 1992, 9:34:39 PM
(...)
15—True Item: President and Mrs. Bush’s tax returns are made public, revealing that in 1991 the President made $2,718 in royalties for his autobiography, whereas First Dog Millie made $889,176 for hers.
 
Google Groups: alt.current-events.clinton.whitewater
Enter Here: “Name the First Dog” Contest Announced by WH
Bill Nalty
Dec 10, 1997, 3:00:00 AM
A New Addition to the White House: a Chocolate Labrador
Young Chocolate Lab in D.C. Needs Your Help
 
Name the First Dog!
ABCNEWS.com
 
President Clinton has a new chocolate labrador puppy named …
 
That’s just the problem. This young dog needs a name. Join thousands of ABCNEWS.com users around the world in contributing possible names for the new first dog. Check back with us and with Good Morning America to see the most popular suggestions.
 
http://www.abcnews.com/sections/us/maildog/
       
Google Groups: alt.english.usage
“Happy New Year”
John Dean
Dec 31, 2002, 1:50:58 PM
I’ll see your superbowl and raise you two rosebowls and a hyperbole. Our cats wrote to Socks and got an autographed picture back and a note where he said he was proud to be the ‘First Cat’
 
Google Groups: rec.sport.tennis
First Dog takes a bite at White House reporter
Dave Hazelwood
Nov 7, 2008, 8:20:18 AM
First Dog takes a bite at White House reporter
Posted: 10:37 PM ET
WASHINGTON (CNN) — President-elect Barack Obama’s daughters have been promised a puppy for the White House — President Bush’s dog, Barney, demonstrated his technique for dealing with the media Thursday…sin
     
Twitter
Jami
@jamiansell
totally saw Beau at the white house!!! #firstdog
10:56 AM · Dec 11, 2009·Twitter SMS
 
Twitter
Christopher Miller
@CScottMillertym
#FirstDog only brings pain, juvy, and no hope of ever finding parents or the president.
12:51 AM · Jul 20, 2011·Twitter for iPhone