Yes we are aware of this, and sadly I don’t think we’re going to be able to match the native experience:
apps running in the browser don’t (currently) have access to the user’s complete set of system preferences, just the language. So the format you will get when inserting a date or time in Bear Web is determined just by the language, assuming no further customisation. For example if the language is set to en-US it will automatically pick the 12 hour format, while en-GB will give you 24 hour format, and so on.