The phrase "specific airline" refers to a particular airline company, not just any airline. It means that the conversation or topic is focused on or related to a particular airline and not all airlines in general. Full definition
I was able to work a few deals with specific airline credit cards and a few other cards to batch them together to get a deal. (debtroundup.com)
Because you are earning miles that are redeemed for travel credit, you don't have to worry about using your miles for specific airlines. (creditcardsmarts.org)
When flying on specific airlines and through airports that have airline specific lounges, even more options are available. (blog.tortugabackpacks.com)