China's Ministry of Foreign Affairs has released several statements saying that it's from China, they know its capabilities and purpose, and that they regret letting it enter US airspace. All in all, it's clear that China agrees that it was a Chinese balloon.
On February 3rd, the day before it was shot down, a spokesperson for the Ministry of Foreign Affairs said during a Q&A:
Q: According to media reports, a Chinese unmanned airship appeared in the US airspace recently. What is China's comment?
A: The airship comes from China and is of a civilian nature, used for scientific research such as meteorology. Affected by the westerly wind and its own control ability is limited, the airship seriously deviated from the scheduled route. China regrets that the airship strayed into the United States due to force majeure.
And on February 5th, the day after it was shot down, the Ministry of Foreign Affairs released a more official statement, part of which says:
The Chinese side has repeatedly informed the US side after verification that the airship is for civilian use and entered the US due to force majeure, which was completely accidental.
Both of those statements are on the Ministry of Foreign Affairs' official site, and I'm relying on an auto-translator to get the English version. Even if some nuance is lost in translation, China doesn't seem to be denying at all that it's a Chinese balloon, and since they claim to know what it was and it was doing it's safe to say the balloon was from China.
How the US knew for sure it was a Chinese balloon likely isn't public information, however according to Pentagon Press Secretary Patrick Ryder and other Department of Defense officials this is not the first such balloon they've tracked so they have some familiarity. Some statements from Ryder and other officials from before the balloon was shot down can be found in this article on the Department of Defense's website:
The United States government has detected and is tracking a high-altitude surveillance balloon that is over the continental United States right now...The U.S. government, to include NORAD, continues to track and monitor it closely.
The official said this is not the first time such a balloon has been seen above the United States, but did say this time the balloon appears to be acting differently than what has been seen in the past.
"It's happened a handful of other times over the past few years, to include before this administration," the official said. "It is appearing to hang out for a longer period of time, this time around...