The nine were all born after Nato's 1999 military campaign, strongly backed by the former UK prime minister, which ousted Serbian forces from the region.
Mr Blair is credited with playing a decisive role in ending former Serbian leader Slobodan Milosevic regime's persecution of ethnic Albanians.
One of those named after the former prime minister, Tonibler Sahiti, said he was a "very great man".
Mr Blair, paying only his second visit to Kosovan capital Pristina since 1999, was greeted by a crowd of several thousand people, some carrying British flags, and posters describing him as a "leader, friend and hero".
He and his wife Cherie appeared on stage with nine children named after him in the aftermath of Nato's air campaign against Serbia.
Before he was introduced to the crowd, Mr Blair was told that his name was "quite common" in the country.
Speaking to Kosovo TV, the father of one of the children named after him remembered when Mr Blair visited a camp in neighbouring Albania where refugees had fled from the violence in their homeland.
"They ask me what the name is," Jahir Sahiti said on his son Tonibler.
"I tell them that the name is Tonibler and I hope that he carries it in good health."
His mother Shukrija said she was very "proud" he was named after Mr Blair, adding: "I hope to God that he grows up to be like Tony Blair or just a fraction like him."
After Nato's action in 1999, it established a UN protectorate over the territory. Kosovo declared independence from Serbia in 2008.
Mr Blair told the audience that he did not regret his actions in supporting the attacks on Serbian troops in what is now Kosovo and targets in Serbia itself.
"I did what was right. I did not regret it then. I do not regret it now," he said.
Ilir Nishku, head of the BBC World Service's Albanian Department, said Mr Blair was regarded as something of a hero in Kosovo due to his championing of the war.
He said he had equal status to the former US President Bill Clinton.
Tony Blair has met nine children named after him during a visit to Kosovo.