Putting on the voice of a news reader, he said: "Why has President Trump given up so much?"
Former South Korean President Kim Dae-jung had also received a pair of Pungsan pups after his landmark summit in Pyongyang with then North Korean leader Kim Jong Il in 2000.
"He wrote me lovely letters", Trump said of Kim during a rally.
"I was really being tough - and so was he".
Trump on Saturday elevated his recent praise of Kim to new heights, at a West Virginia rally in support of local candidates for his Republican Party.
He then added: "I didn't give up anything".
Before they turned the page on decades of public acrimony, the leaders regularly traded threats and insults as North Korea pushed to develop a nuclear missile capable of hitting the United States.
2m-high tsunami hits Indonesia after strong earthquakes
He said the fate of "tens to hundreds" of people involved in a beach festival in Palu when the tsunami struck was unknown. Indonesian officials issued a tsunami warning Friday after a powerful natural disaster rocked the island of Sulawesi.
But denuclearization negotiations have stalled.
Somewhere the middle of his speech, after suggesting the USA was about to go to war with the North Korea when he took office, Trump waxed nostalgic about how he and Kim exchanged threats to annihilate each other with nuclear weapons, then claimed it was Kim's "beautiful letters" that turned everything around.
'We were going to war with North Korea. Millions of people would have been killed.
Speaking at the United Nations General Assembly on Saturday, North Korea's top diplomat said the nation won't dismantle its nuclear weapons until it has "sufficient trust" in the USA, and called on the Trump administration to drop its "coercive methods" such as sanctions.
He claimed his growing closeness to Mr Kim has resulted in the end of rocket tests, helped free hostages and is getting the remains of American servicemen returned home.
Trump followed those comments by saying Wednesday he had received an "extraordinary letter" from Kim, and sounded optimistic about prospects for a second summit between the two leaders "fairly quickly".
Kim, after a third meeting with South Korean President Moon Jae-in Kim this month, said he would dismantle North Korea's main nuclear complex and accept worldwide inspectors at a key missile site in exchange for unspecified corresponding measures from the United States. Instead, he noted, the U.S.is continuing sanctions aimed at keeping up pressure.