Beerus and Champa's home planet are one of Dragon Ball Super's biggest mysteries. While the two are based on a species of hairless cat, where they come from and if there are more of their species hasn't been made clear yet -- but fans are speculating that Dyspo from Universe 11 may be of their ilk, given his resemblance to Beerus. Even Champa made note of the relation. Unfortunately for fans, that was the only time the series has brought it up so far.
Dyspo's design is based more on a rabbit, as his ears are much longer that Beerus' and he doesn't have a tail like the God of Destruction. He also appears to be hairless like Beerus and Champa. Like the two gods, his origin is unknown, but he has a strong sense of justice, which is more what can be said for Beerus. His fighting style is based on speed, and even Champa had a hard time seeing Dyspo when watching him fight.
Beerus was originally supposed to be a lizard, but Toriyama decided to change the design to be based on his Cornish Rex cat, Debo. Being 14 years old, the cat fell ill at some point, but lived despite the vet saying he didn't have much longer, prompting the vet to call Debo a "demon." He was also meant to be more of an antagonist and have the ability to infect others with evil. His name even a play on the Japanese pronunciation of the word "virus."
So what's the connection? Fans have been taking to places like Quora to try and figure out why one of the contestants of the Tournament of Power looks just like a God of Destruction. Youtuber ExperGamez suggests that it's likely more of recycling designs by staff than it is a major origin point. He points out that it wouldn't be the first time that the series has reused designs, and cites Recoome from the Ginyu Force as an example. Recoome looked just like a human from Earth, but was from some other species and planet. It isn't out of the realm of possibility.
But, it could also be that Beerus, Champa and Dyspo come from the same planet, where everyone is an anthropomorphic, hairless animal that has purple skin and large ears. Given the similarities between the two, it could be that Dyspo represents what Beerus could have been had he not been a God of Destruction. Beerus doesn't care about the fate of others, as seen when he refused to help during the destruction of the Saiyan race.
Dyspo, on the other hand, is more caring and has the goal of ridding his universe of evil, joining the Pride Troopers to that end. Since Beerus was supposed to originally spread evil, it could be that Dyspo is supposed to be his polar opposite. They are two sides of the same coin. However, until Toriyama or someone on the production team comes out or the series goes into the backstory for Champa and Beerus, fans may just have to keep on speculating.