Is there a way to set the styling for the "choose file" button for Webform module?
Yes, actually styling a FILE INPUT field is hard because, just like you're seeing, it's a singular field that is hard to pick apart.

Here is a discussion of styling file input fields from stackoverflow.

As you can see - thought there are many solutions presented, they all require some jquery and essentially hide the input field and replace it with something else.

So in short - there is no easy way to do it.