I am very confused as to this idea, of a solder returning from probably WW-I (1917 ad!), buying a swagger stick for his little girl.

Would a single little girl in 1917 prefer a swagger stick to the dolls or the pinwheel? Swagger sticks are called out in the biggest font, the very centerpiece of the ad, but they don't even bother to show a swagger stick? Was this hyper progressive, at 50+ years before the army let women become officers? Did somehow, the Marks Brothers Toy Company create a branch in the timeline where that happened decades before the other timeline where no one ever imagined swagger sticks as gifts for little girls? Maybe we'd still be waiting on female generals and admirals today, in the other timeline?

I have so many questions.

And what's a swagger stick?

Edit: Oh now I get your comment.

Yup, that's it! Officers were encouraged to carry them as a symbol of their authority.