This is an interesting phenomena, which could need some help also from developers to potentially help Grammarly developers to figure out how Obsidian could be also supported on Windows. Or if it will be at all possible due implementation differences on separate platforms.
I am using paid version of Grammarly both on OS X and on Windows.
Grammarly works on OS X version just fine, but on Windows the floating widget never appears.
I have noticed few discussions about the topic previously - especially regarding the plugin system that was a workaround for some time. I opened up also a ticket to Grammarly to ask about the issue, and could link this conversation to the ticket if there is need / want for support.