Godbolt says Compiler Explorer right there, the fact people call it "Godbolt" is because that's the memorable name rather than because Matt Godbolt tried to name it after himself. He calls it Compiler Explorer.

You can't control what people call things. My favourite example is Blackfriars Bridge in London. When that bridge was built, it was formally named after William Pitt. But like, who cares? In what way is this bridge, Pitt's bridge? People knew where it was, the Blackfriars area of London, so too bad William Pitt, everybody called it Blackfriars Bridge. And so, that's what its name is.

The priory which is why "Blackfriars" got the name hadn't existed for two hundred years by the time the bridge was proposed. Nobody living when the bridge was finished would have ever known anybody who'd met any of the friars during their lifetime. Nevertheless that part of London had "always" been called Blackfriars, and so a bridge and a railway station built many years later are named that too.

People call it Godbolt because you access it via godbolt.org,