Born in Chesterfield, Missouri, Kunin had a number of NHL players for coaches during his minor ice hockey career, including Keith Tkachuk, whose son Matthew he played alongside as a child.

After spending time on a Minnesota line with Jordan Greenway and Joel Eriksson Ek, Kunin was traded to the Predators prior to the 2020–21 season.

After a slow start to the year, he finished on a scoring streak, including a double-overtime victory over the Carolina Hurricanes in the 2021 Stanley Cup playoffs.

[7] Kunin also leaned on his older brother Nick, whose own hockey career was cut short by a series of concussions, for support as he became accustomed to managing his condition.

[9] As he had finished high school early, Kunin entered the program as the youngest skater on the team, but he impressed coach Mike Eaves with his maturity, skating skill, and shooting abilities.

Kunin once again led all Badgers in scoring with 22 goals and 38 points, netting him an All-Big Ten Second Team placement, while Wisconsin won 12 more games that season than the last.

[21] Kunin was one of the last players cut from the Wild roster before the 2017–18 season, his place in the lineup effectively taken by free agent signing Matt Cullen.

[22] When a number of injuries befell the Wild roster, however, Kunin made his NHL debut on short notice,[23] skating in less than 14 minutes of the home opener against the Columbus Blue Jackets before he was sent back down to save space on the team's salary cap.

[26] Kunin was once again called up to Minnesota on February 27 in order to give the team more energy as they finished out the season, but on March 4, during a game against the Detroit Red Wings, he suffered a season-ending anterior cruciate ligament injury in his right knee.

[29] Kunin was medically cleared to begin playing hockey again on October 8, 2018, and was assigned to Iowa to continue his recovery and spend more time on the ice before rejoining the NHL for the 2018–19 season.

[31] He was meant to return to Iowa a week later, but an injury to Matt Dumba forced coach Bruce Boudreau to rearrange his offensive lines, and Kunin remained in Minnesota skating alongside Koivu and Jason Zucker.

[32] He was briefly sent back down to Iowa at the end of January to play in two AHL games while Minnesota had an eight-day break,[33] and he was recalled to the NHL on February 4.

[37] After a difficult start to the 2019–20 season, the Wild found a stable, productive line by mid-November in Kunin, Jordan Greenway, and Joel Eriksson Ek, affectionately referred to as the "GEEK Squad".

The Wild had clinched a spot in the 2020 Stanley Cup playoffs, with Kunin invited to participate, but he was initially hesitant to break quarantine, as his diabetes diagnosis placed him at higher risk of suffering complications from the virus.

[49] He also scored six goals in the last nine games of the regular season, including two in the Predators' 3–1 defeat of the Carolina Hurricanes on May 8, which allowed Nashville to clinch a berth in the 2021 Stanley Cup playoffs.

[54] On December 4, 2021, Kunin's 24th birthday, he recorded a Gordie Howe hat trick, the first for Nashville since Ryan Johansen three years prior, en route to a 4–3 overtime win against the Montreal Canadiens.

[61] Early in the first period of the Sharks' December 13 game against the Arizona Coyotes, Kunin left the ice following an awkward hit on Patrik Nemeth.

[64] Early in the season, he found chemistry with linemates Ryan Carpenter and Givani Smith, the trio combining for a six-point night against the St. Louis Blues on November 16.

He was promoted to the second line, and general manager Mike Grier saw Kunin as part of a veteran leadership group who would help guide younger players in the organization.

Serving as an alternate captain for the gold medal-winning team, Kunin scored two goals and recorded six points en route to the championships.

The following year, he scored six goals and served as captain for the United States for another gold medal performance, this time at the 2015 IIHF World U18 Championships.

[73] Kunin made his senior international debut at the 2019 IIHF World Championship in Slovakia, playing on a line alongside fellow NHL skaters Luke Glendening of the Detroit Red Wings and Frank Vatrano of the Florida Panthers.