With the new ringer button on iPhone 15, how can I tell if the ringer is on or the phone is on silent?

If the slashed bell icon does not show up, then it's not silenced.

If you have the side button set to silence the iPhone, pressing and holding down on it will cause an animation to appear telling you its been silenced and the icon will then appear and stay there until you press and hold the button again. When you press it again another animation will appear and state its been set to "Ring".

What happens when you press and hold the button now?

If you have the button set to something else, then it will perform that action. You can see what its set to in Settings ➜ Action Button

If you see an icon of a bell with a line through it at the top of your screen, Silent mode is on. For this and other status icon meanings, see:

Status icons and symbols on your iPhone - Apple Support

Regards.

Sounds like it's not set to silence then. Or it's not actually working correctly.

If you go into Settings ➜ Action Button what is it set to do?

You can scroll left and right to select an action for the button to perform when held down.

You're welcome. Glad it worked out.