Why is my neighbor receiving AirTag notifications from my golf bag's AirTag?

To put it simply, any iPhone within Bluetooth range (~30-40') of your AirTag, AND stay within that range will eventually get this notification. As I'm sure you read in the support article, this is deliberate design feature to prevent these tags from tracking someone.

A similar scenario, would be if you travel with a friend in the same car and you have an AirTag attached to your car keys, your friend may get this same notification. Although a bit annoying, think of the case where you are driving and someone plants a tag somewhere in your car. This is definitely when you will want to get the notification so you can find and disable the tag.

It should only show up if they are together for a period of time. So just being next to luggage with someone else's AirTag wouldn't necessarily show up on your device unless it was with you for a period of time.

Here is an Apple support document that describes what is going on. What to do if you get an alert that an AirTag, set of AirPods, Find My network accessory, or compatible Bluetooth location-tracking device is with you - Apple Support

My guess is that if the two of you were together playing golf, it is possible that their phone was close enough to your bag with the AirTag that it felt it was following them. What your neighbor saw was what they would see if they had your AirTag in their bag. I suspect it had to do with proximity.