What if they're just selling other stuff like food, clothes,... not games?
The other company's ability and desire to "go after you" is highly variable. The bigger you become, they more successful you are, that just makes it more likely your startup will become the target.
As an example, if you are just a single individual with an unvisited website running with a name like "Target Games" and a bullseye logo, chances are good Target Brands will not notice you nor care. But as you grow, and start to rise in the search engines, and your name starts to show up in search results next to Target's commercial store offerings, you can begin to count the days until you will be asked to sign to acknowledge receipt of a Cease and Desist letter.
As far as that goes, SeanMiddleditch answered it clearly:
You're trying to be a developer in a highly creative industry. Being creative with a company name is the absofreakinlutely easily thing you should be able to do.
Are you a creative person in a creative field, or not? Even if you are only slightly creative you can come up with unique and novel names. The pattern "Adjective Animal" (e.g. "Naughty Dog", "Red Ant", "Big Fish", "Buzz Monkey"...) was popular for several years, including spawning a company with that actual name "Adjective Animal".
Feeling particularly unoriginal? Visit a random word generator, keep picking random "adjective noun" pairs until you find words you love.