No. Each active instance (whether cloud or self-hosted) requires its own license. However, one license can cover multiple technicians if you purchase the "multi-technician" add-on.
Store the key in a password manager. If you lose it, contact ConnectWise support with your invoice number. Screenconnect License Key
Before discussing keys, you must understand that ScreenConnect abandoned the traditional "perpetual license key" model several years ago. Today, it operates on a SaaS subscription model (cloud-hosted) or a self-hosted annual subscription. Before discussing keys