I checked the officials newsfeeds on bybits and also their social media handles, I did not see any where that the official bybits announce the lunch of their telegram games, and from the m the link you shared I can't fine any link between bybit exchange with the so called bybit space that your game carries.
Me too I received a lot of email promotion on Bybit but never an invitation or announcement about their new game, Checking the game, they are using Bybit logo but theren's o way to connect the application to your Bybit account just liek what Binance and OKX asked to their users on their launched Telegram based airdrop.
So is either you have to use bybit name just to get the attention of the cryptocurrency community, and using them to lure players, which is not an honorable thing to do as a new brand, because no body will take your game platform serious if you have to steal build on the identity of others.
The game was launched October 7, so if this is really Bybit owned they should have announced this, it will be a month since the launch, and the memberships is not that huge when Bybit has millions of members, I believe they are riding on Bybit popularity I believe will eventually make an announcement if this becomes viral in the airdrop community.