That’s not how it works. Trademarks are limited to the area you operate in.
So I could open up a company called SuperDry that makes cat litter and neither would be able to sue me because their protections are limited to fashion and brewing/drinks respectively.
The reason they’re taking action against City is because they think the club is selling clothing with SuperDry on and the clothing company’s trademark is in that category.
Even if they win the consequences would likely be just be that us fans get to buy training kit etc without the sponsor. The players would still wear the Asahi sponsored kit.