Achievement Pictures on the notification like other platforms

It depends on the game. Many times it's one set of achievements across platforms, usually with the Play Anywhere titles, but not all. Games like Forza Horizon 4 have a shared list across platforms, while games like Minecraft have discrete achievements
per platform. Additionally there are games available on Windows 10, Android, or iOS that have achievements that aren't on other platforms, like Forza Street for Windows 10 (coming to mobile soon) and Wordament for iOS and Android.

Are there specific games you're curious about?

So it's basically as of right now a picture of whatever it is (specific achievement pic, gamer pic, game icon, etc.) then a icon of what type of notification (SmartMatch, Achievement, Download Complete, etc.) and text with the announcement (achievement unlocked,
match found, etc.) in bold and details below (how much gamerscore, maybe achievement description, who the match is with, etc.).

Thats what I took from it. It looks a lot nicer than 360's notifications which is good.

I wouldn't assume that until we see it. A big complaint about the 360 notifications is that the same notification pops up for messages, invites, people coming line, and achievements. I wouldn't be surprised if they looked into differentiating them, if not
at least slightly. I can see what you mean though, maybe have the achievement picture where that game tile is, then have the description next to it. We don't know though yet

Hi, I'm Luis a Xbox user just like you. Thanks for reaching out us here.

To disable achievement notifications:

Setting>Preferences >Notifications >Xbox notifications >look for achievements >Press X to toggle on/off

Hello Stelardag, this is Manuela. I'm an Independent Advisor and community user like you, ready to assist with this.
Not a Microsoft employee.

I’m sorry to hear you’re having trouble with the achievements' notification. Let’s try to sort this out.

Please review if achievement notifications are enabled on your console by following the steps below:

Open Settings > Preferences > Notifications > Xbox Notifications > Achievement notifications
Make sure everything is enabled except the third option.

Lastly, I would like to ask, is your console enrolled in the Xbox Insider program?