When a guest books on your website, the checkout validates their card with Stripe and saves it as a payment method. It does not charge the card.
Who charges the card. Your PMS handles the charge after the reservation is confirmed, according to your PMS payment automation settings.
If a manual charge from your PMS is the only charge you see in Stripe, the system is working as designed. There is no risk of a double charge between the website and your PMS.