I believe ladies suspender-belts are called 'garter-belts' in the USA. They were not used to hold up men's trousers. The British call such devices 'braces.'
Suspenders are small straps or bands that hold up socks or stockings. They are straps that run over the shoulders and support or hold up trousers or a skirt.
One of two supporting bands worn across the shoulders to support trousers.
attached to the front and back of trousers and held up by the shoulders to keep pants in place
(shoulder straps to hold up men's pants) braces
Suspenders, braces, garters, and garter belts are clothing accessories.