How do I put Face ID an a certain app?

you can't. The app developer needs to program the app to use it. So if it doesn't show up in Settings > [app name] > Face ID, then you won't be able to use Face ID for that particular app.

You can contact the app developer and suggest integrating the feature

How to contact an app developer - Apple Support

No you cannot.

Are you sure that he didn't just unlock the phone using Face ID and the stocks app was open (because he locked his phone with the app in focus)?

I don't know how he is doing that..

maybe someone else will comment in

Are you referring to the native Stocks app on the iPhone, or another app? Once you have unlocked the iPhone, the Stocks app does not require anything to open it.