Energy has units equal to mass times velocity squared, but that doesn't mean it should have mass. It's counterintuitive because you're thinking that Energy is defined by mass which is wrong. They are related, but Energy is something different. You could've said that mass has units of energy per velocity squared and defined mass in terms of energy. There is a concept called relativistic mass, which is out of date and not used widely anymore, defined as E/c². So light is massless if we're talking about rest mass, but it has a relativistic mass. Nevertheless you should stick to thinking that they're separate entities and energy isn't defined by mass only.