Great. Does this mean we can wait to get one day a labels dropdown menu when clicking on a page watching star, alongside with the dates menu that is there now?

Yep, that's the goal of T415173 (this task is just a part of it, and we've not yet made the other tasks that make up the whole feature).

Change #1238472 had a related patch set uploaded (by Samtar; author: Samtar):

[mediawiki/core@master] ApiWatch: Update action=watch API to accept label IDs

https://gerrit.wikimedia.org/r/1238472

Request

{
	"action": "watch",
	"format": "json",
	"labels": "1|2",
	"titles": "Main Page",
	"formatversion": "2"
}

Response

{
    "batchcomplete": true,
    "watch": [
        {
            "title": "Main Page",
            "ns": 0,
            "watched": true,
            "labels": [
                {
                    "id": 1,
                    "name": "test"
                },
                {
                    "id": 2,
                    "name": "two"
                }
            ]
        }
    ],

labels here will contain the resultant labels applied to the page (not just the labels added in the request)

Change #1238472 merged by jenkins-bot:

[mediawiki/core@master] ApiWatch: Update action=watch API to accept label IDs

https://gerrit.wikimedia.org/r/1238472

Change #1240679 had a related patch set uploaded (by SomeRandomDeveloper; author: SomeRandomDeveloper):

[mediawiki/core@master] ApiWatch: Use a batch query to load watchlist labels

https://gerrit.wikimedia.org/r/1240679

Change #1240679 merged by jenkins-bot:

[mediawiki/core@master] ApiWatch: Use a batch query to load watchlist labels

https://gerrit.wikimedia.org/r/1240679

Thanks, tried, and confirmed the question I was worried about, that the action does not erase the notificationtimestamp value as usually at watch actions. Great job!