https://github.com/godotengine/godot/releases/tag/4.3-stable
Godot 4.3 is a feature release improving upon the previous version in many aspects, such as usability and performance. Feature releases also contain new features, but preserve compatibility with pr...
Latest I've heard the necessary WASM APIs are planned for .NET 9, which is targeting November. Godot team previously said they'd jump into it once those were available. Long story short, not in 4.3.
I don't keep up with unity much, but I do know they were using Mono for the longest time, which wouldn't have the same restrictions as the newer "Core" runtimes. I think their efforts to catch up were called CoreCLR, that might be a lead to how their progress is.
Edit - some research and to explicitly answer your question: maybe. The unity team has built a custom compiler and bindings to bridge the gap between their APIs and newer .NET versions. They're essentially supporting parallel build/export pipelines, while Godot is trying to keep it simple and inline with what Microsoft provides.