No. There was literally just a box to type a name in and a “rename” button. Not any type of “file save” dialog box which lets you select a drive and path, etc.
This seems to be a 8.1 “feature.” See the following, for example:
http://techjia.com/where-are-files-stored-for-windows-sound-recorder-windows-8-1/
I don’t have 8.1 handy, I tried 8. Sounds like they busted it, appdata should at best be where the temp file is stored, user data should never go there. And now we see why they didn’t change sound recorder for 20 years, first time they changed it they broke it.