I'm assuming you're taking your phone with you. Is your phone set to be your location? (This is the 'My Location' setting on the Me tab in Find My.) This setting is how it detects when you're away from your AirTags. If you have something else set to be your location, such as an iPad that always stays home, then you wouldn't get the notification because it thinks you're still home with the AirTags.
This may just be a semantics thing, but to be clear...the AirTags aren't linked to a device; they are linked to your Apple ID. Do you and your wife happen to share an Apple ID, or is it indeed linked to her own Apple ID? If you share an account, then it could be related to my previous comment above due how your Apple ID's location is detected. If they're separate, is your wife able to see the dog's location when you take it for a walk?
Otherwise for both cases, have you given it enough time for it to notify you? I've noticed it sometimes can take five minutes or more, and I'm already at least a block away.
Other things to check: Make sure you have Location services enabled, Bluetooth turned on, and your phone is participating in the Find My Nework (Settings -> Apple ID -> Find My -> Find My iPhone -> Find My network). This allows the phone to detect nearby AirTags and relay their location to Apple's Find My Network.