Several airlines offer birthday cakes, or at least did before the pandemic (many catering options and special meals were discontinued during COVID).
Turkish Airlines offered a "celebration cake," though the program is currently suspended. When they were being offered, they'd provide free cake for birthdays and wedding anniversaries. You can see some pictures of those who received it.
This 2018 article lists other airlines that have offered birthday cakes: "Korean Air, Turkish Airlines, SpiceJet, Swiss, Qatar Airways, and Cathay Pacific are just a few airlines that will bake a cake or special dessert for onboard enjoyment." Some other airlines may offer a special treat to top-tier frequent flyers reaching certain milestones. It looks like Indian carriers GoAir, SpiceJet, and Vistara have offered cakes, and IndiGo used to.
Some airlines charge for the service: Hong Kong Airlines has several cake and champagne options, and Cathay Pacific had a charge as well. Emirates used to before the pandemic. And some airlines (apparently including Singapore) might be able to offer one but don't advertise it as an option. Cakes may only be available on certain routes.
The pandemic makes an answer with a complete list of airlines not really practical, since it seems many airlines that previously had cakes aren't doing them right now. But a search for "[airline name] birthday cake" should provide any relevant information.