What does 'airline approved' actually mean for a cat carrier?

Specialist community consensus (notably r/CatAdvice) is blunt: there is no universal TSA 'approval' for carriers. Airlines—not TSA—set under-seat size limits, and those limits vary by carrier and even by aircraft. Reviewers consistently advise measuring the under-seat space for your specific airline and route, then buying a soft-sided carrier that can compress to fit, rather than trusting a listing's 'airline approved' badge.