Mattel began in 1945, as a garage workshop owned by Harold Matson and Elliot Handler. Their business name "Mattel" was a combination of the letters of their last and first names, respectively. Matson soon sold his share of the company, and the Handlers, Ruth and Elliot, took full control.

Mattel's first products were picture frames, however, Elliot started making dollhouse furniture from picture frame scraps. That proved to be such a success that Mattel switched to making nothing but toys. Mattel's first big-seller was the "Uke-a-doodle", a toy Ukulele. It was the first in line of musical toys. The design went on to be used on plastic "music boxes" called Strum-Fun Getar This could be played automatically using a disk or it could be tuned and played properly(?) 

They went on to make a lot of the Disney Ukulele's of the 1950's and 60's, as well as Barbie. The don't appear to be making Ukuleles at the moment; unless you count the one in Ken's Hawaiian adventure? 

The Uke-a-doodle design was apparently not only used by Mattel. In the 40's, early Carnival Ukuleles use it, as does the non Mattel Uke-a-Tune Made in U.S.A. I don't know if Mattel made them for others or if there was another plastic factory making them for Mattel and the other people too?