AirTag isn't showing among my devices

Airtag is an item and not a device.

People, Devices and Items under Find My: Here's a clarification:

1. People/Devices: You can share your location with others using the Find My app. You can choose to share your location temporarily or continuously with specific contacts. This feature is typically used for safety or convenience, such as when meeting up with friends or family.
2. Items: With the introduction of AirTags, Apple allows users to track physical items like keys, wallets, or bags using the Find My app. The key difference here is that the location of items is not meant to be shared with others by default. The primary purpose is to help the owner locate their lost items. However, if someone else's item is near your iPhone or Apple device with Bluetooth and Location Services enabled, you might receive a notification about the nearby item. This is a privacy feature to prevent unwanted tracking.

See the pic below