Most... but not all... of the all-metal Japanese zigzag sewing machines like this one have a "JA" number stamped into the metal on the underside. Sometimes they also have a tag or stamp with a "JC" number. An example of this might be JA-8.
*Be careful pivoting it back to look on the underside. Machines often like to scoot when the center of gravity changes like that. May be best to do this on a carpet...
The JA or JC number might be able to at least give you a manufacturer's name. If you have that much, you could search for other machines made by that company and perhaps find similar sewing machines. It wasn't uncommon to see several different badges (or names) on the exact same model of machine.
To find the best and most updated list of JA/JC numbers online try:
this link.