Correctly identifying the object is very dependent on knowing its exact size -- but you didn't tell us its size, despite instructions to do so, posted at the top of the What-Is-It? forum.
In your photos, there's no way for use to know the object's size. It could be 3/4-inch wide or 3 inches wide or anywhere in between.
Here's why your object's size is important. Suspenders straps tend to be wide and thick fabric. Garter straps tend to be narrower and thinner fabric than suspenders straps. The difference is because not much strength is required for holding up a sock or stocking, but much more strength is needed to hold up a pair of pants. Therefore, the buckles/clips/clasps are "heavy-duty" for suspenders straps and "light-duty" for sock/stocking garter straps.
Your object has a small tongue-shaped tab ...a feature which is seen on sock/strocking garter strap buckles. The little tongue-shaped tab's purpose was to hold up a cord -- NOT to grip another corresponding strap-end buckle. Examine the photo below, showing an early-1900s advertisement for a sock/stocking garter. Note, the part I'm talking about is not gripping the sock -- I'm talking about the part which holds the cord. In the advertisement's illustration, you can see the little tongue-shaped tab holding the cord.
If your object is narrow, say about 1-inch to 1.5-inch wide, it is more probably from a sock/stocking garter than from suspenders.