You actually don't write any data, you just create a directory. It might be that the userfs only stores actual files (with their path). Just guessing though.
Try to open and write a file.
I can write to user:// in HTML5 and I can see the data in the (Firefox) Inspector under Indexed DB -> -> /userfs -> FILEDATA -> one key per file name. I didn't check isuserfs_persistent though.
Here's my app if you want to check if your browser is compatible with Godot and Indexed DB:
Just change something in the game options and data should be written to Indexed DB.