If you’re showing both sides of it, I’m not convinced it’s a button. It seems to have designs on both sides, and no sign of where an attachment loop might have been.
In your first picture at 9 o’clock, I can see the lower half of a letter ‘E’, above the outline of what looks like the top of an armorial, and another letter with a single upright (or possibly an 'N') the other side of it:
View attachment 2078327
In your second picture, on the lower half at 7 o’clock I can see what looks like the outline of a large arrow pointing upwards, with perhaps a double-stemmed letter or ‘Roman’ numeral ‘II’ in the pointed portion:
View attachment 2078328
[Both pictures enhanced and rotated for clarity]