Miriam - if you take some measurements on the feet, I'm almost sure I can tell you who made your machine. I've found that these are rather unique for each Mfg no mater what the machine looks like.
See if you can swap out the plates on the Regal and the "no name". I know one will be shorter than the other, but see if they fit otherwise. This includes having the round cutout in the same spot. I have some that are the same width but the circle isn't in the same spot. I'd lean that they are from the same Mfg - and that it's likely National. I've not encountered any other machine that takes the same attachments as National.
1. the gap on the foot is wider, closer to 3/16 than 1/8 and the distance from the presser bar and needle will be vey close to 1/2" - and the needle is centered in front of the presser bar.
I must have machine by at least 15 different Mfg at this point - and other then those made to be the same as Singer - they all have something different about them for the attachments.