“A guy walks into a bar…”  is a typical form of what has been called the “bar joke.” A newt version is:
 
A man walks into a bar with a newt on his shoulder.
The barman says, “That’s a very nice newt. What’s it called?”
The man says, “Tiny.”
The barman asks, “Why’s he called that then?”
The man says, “Because he’s my newt (minute).”
 
The joke was posted to the newsgroup alt.pro-wrestling.wwf on December 30, 2000.
 
         
Google Groups: alt.pro-wrestling.wwf
JAS!!
Fallen
12/30/00
(...)
A man walks into a bar with a newt on his shoulder.
The barman says, ‘Thats a very nice newt, what’s it called?’
The man says ‘Tiny’.
The barman asks ‘Why’s he called that then?’
The man says ‘Because he’s my newt’   
 
Google Groups: alt.fifty-plus.friends
A tiny joke
Fair Dinkum
10/27/01
A woman walks into a bar with a newt on her shoulder.
The barman says “You’ve got a newt on your shoulder, love. What do you call him?”
The lady replies “His name is tiny”
The barman replies “Why Tiny?”
The woman replies “Because he is my-newt”
 
31 March 2002, Mail on Sunday (London, UK), “Jokes,” pg. 73:
A woman walked into a bar with a newt perched on her shoulder. She ordered a drink for herself and one for the newt. ‘What’s his name?’ asked the bartender.
‘Tiny,’ said the woman.
‘Why Tiny?’ he asked.
She replied: ‘Because he’s my newt.’
 
Google Books
A Prairie Home Companion
Pretty Good Joke Book
By Brian Becker
Minneapolis, MN: HighBridge
2003
Pg. 130:
A man walks into a bar with a newt on his shoulder. He told the bartender that the newt’s name was Tiny.
“Why?” asks the bartender.
“Because he’s my newt!”
 
17 April 2007, Liverpool (UK) Echo, “Time Off: Your Jokes,” pg. 22:
A man walks into a bar with a newt on his shoulder. The barman looks at the creature and asks the man what he calls it.
“Tiny” replies the man.
“Why’s that?” asks the bartender.
“Because he’s my newt!”
 
Twitter
Vikram
‏@V1kram
A guy walks into a bar with a newt on his shoulder. “What do you call that?”, asks the bartender. “I call him Tiny, because he’s my newt!”
12:43 AM - 8 Jan 2008
   
Google Books
The Mammoth Book of Really Silly Jokes:
Humour for the whole family
Edited by Geoff Tibballs
London: Constable & Robinson Ltd
2011
Pg. ?:
A man walks into a bar with a newt on his shoulder.
“Hey,” the bartender says. “What’s his name?”
“Tiny,” the man replies.
“Why call him that?” the bartender asks.
“Because he’s my newt.”
 
15 January 2016, llawarra Mercury  (Wollongong, New South Wales), “Calm the heck down, people it’s funny” by Julian O’Brien, pg. ?:
Barry walks into a bar with a newt on his shoulder.
The barmaid looks at the creature and asks the man what he calls it.
‘Tiny,” answers Barry.
“Why’s that” enquires the barmaid.
“Because he’s my newt’ concludes Barry.
 
Twitter
Crap Jokes
‏@TheCrapJoker
I walked into a bar with my pet newt. The bartender asked ‘What’s that thing? Its tiny’
‘I know, Its my newt’
2:20 PM - 5 May 2015