Why was I given a quest from Chapter 5 during Chapter 1?

After choosing a house (Black Eagles) during my first playthrough of Fire Emblem: Three Houses, I got a strange notification:

A quest has been added from Chapter 5 of the main story on. You will find the quest via the bulletin board. In order to accept it, you must have the ability to invite people to tea parties.

The bulletin board only lists three quests: Fresh Catch, Green Thumb Beginnings, and Share a Bite. None of them are labeled as being from Chapter 5.

Why was I given a quest from Chapter 5 while I’m still in Chapter 1? Is it a quest I can work on early?

After some digging, and while difficult to determine as it stands, it's a good chance the quest the notification is referring to is Esteemed Company.

This guide contains a running list of patch notes and updates, and one of the notes for v1.2.0 states:

The additional quest will become available from Ep. 5, Tower of Black Winds, onward as long as Tea Parties have been unlocked.

The wiki's quest lists states this quest as added in v1.2.0, so those pieces of information match up.

Why was I given a quest from Chapter 5 while I’m still in Chapter 1?

Based on the information above, you were not actually given the quest yet. Given the fact it is your first playthrough, it is merely notifying you that a new quest is available in chapter 5 after meeting the certain requirements. When you accept it, then it will appear on the bulletin board.

When v1.2.0 was released, I was well into my 3rd playthrough and never received this notification since I was on a NG+.

Is it a quest I can work on early?

This is not a quest you can do early. You do not unlock Tea Parties until Chapter 4, and Esteemed Company unlocks the ability to invite Rhea to Tea Parties. At best, you can focus building up your support rank with Rhea, but that wouldn't affect very much