Critik asked, about the suspenders strap-adjuster buckle component:
> So how did this stay connected to the other piece I do not see anything broken on it.
I somehow posted a photo showing a similar suspenders-buckle's front instead of the back which I intended. I'll do that at the end of the post, so you can see the sawtoothed-bar on it.
How the sawtoothed-bar was connected to the rest of the buckle:
The second photo below shows a flange with a round hole through it on each side of the buckle. Your sawtoothed-bar's ends have a short tab-shaped projection, which fitted semi-loosely into the round holes, thereby allowing the bar to rotate.
> Is there a name for the fancy on it?
I suspect there is a name for that design, but I don't know it.
> Is that a common used fancy.
Yes. It appears as a decoration on many-many items made throughout the 1800s, and is still seen occasionally on "modern" items. A similar "fancy" is on the tongue of the second suspenders-buckle shown below.