I turn on my homepod mini and it doesn't recognize me

Make sure Voice Recognition is ON

On your iPhone:

Home → (Your Home) → Settings → People → [Your Name] → Recognize My Voice → ON

If this is off, HomePod will treat you like a guest.

Also turn on:

Siri & Search → Listen for “Hey Siri” → ON

Siri & Search → Allow Siri When Locked → ON