Found this out by accident while playing AC Syndicate today. I was having some internet problems earlier this afternoon and decided to play a little, and because of the internet, I had to play in offline mode. During this time, I unlocked a few achievements for the Jack the Ripper dlc, and lo and behold, they actually showed up on the corner of my screen as a notification. Sometime later I remembered I was still in offline mode, and after doing some testing (reconecting to the internet, playing the game again and unlocking one achievement), I noticed the pop-ups were gone again. So, yeah, they still exist in the launcher's code somewhere, but only work on offline mode.
Here's the funny part though: when I was offline and ingame, hitting Shift+F2 brought up the launcher's overlay and it looked like the old version of it (back when it was still uPlay). Of course there wasn't much to see except an error message telling me I was offline, but still, remember that part about it looking like the old overlay.
When I reconected to the internet and played the game again, I hit Shift+F2, and while I got the current overlay, I realised something: The achievement section is nowhere to be seen.
I'm not a programmer or anything, but from my point of view, it looks like they just... Forgot to add achievements to the code of the new overlay, and that's why there's no pop-up for when you unlock one: Because this function literally does not exist anymore. Thanks to what is likely an oversight, the old launcher's overlay's code is still there, and is only accessible when you're offline, hence the pop-ups function normally when doing so.
If I'm right about this, this is the most stupid reason in the world as to why achievement notifications haven't been working properly for the past 1-2 years. My god, Ubisoft, hire some people to fix this horrendous launcher, I beg of you.