How do I put my iPhone 16 in discovery mode?

If you mean bluetooth, open settings and tap on bluetooth. that's it. It's discoverable as long as you're on that page.

Keep in mind that you can NOT pair it to a computer, phone or tablet. That has never been supported.

Bluetooth pairing is not required for features like handoff, continuity, air drop, etc.

BKHAL wrote:

I’m trying to connect a speaker but it isn’t showing up

Here is how to: Connect iPhone to Bluetooth accessories - Apple Support

If that sequence is not working, and the device is in its pairing mode, try pairing this iPhone with a different Bluetooth device, and try pairing this Bluetooth device with a different computer.

If the device is malfunctioning, check with the seller or the with the device support organization.

What exactly are you trying to do?

You need to put the speaker into discoverable (pairing) mode.