No the gum that you swallow will not get stuck in your stomach.
It's an old wives tale that swallowed gum stays in your stomach for 7 years or longer and that doesn't really happen.
Your body will eventually just send the gum out of your stomach and into your intestines where you'll poop it out even if you do not notice the gum in your poop.
The same is true with any other hard object such as a coin, etc the coin would stay in your stomach for a bit longer and then your stomach would decide it cannot digest it so then the coin will be moved out of your stomach and into your intestines.
Then it'll leave your body through your poop.