I prefer xbox achievement because of the achievement value.  Though I do see the advantage of a platinum within the playstation ecosystem. 

In game achievements. At least, I can get a tangible reward in game that helps me with in game activities.

1. PlayStation Trophies: I really like the distinction of bronze, silver, gold, and platinum. I also like how they take a screenshot on PS4 and a screenshot and video clip on PS5. I do have to say though that PlayStation admittedly has too much EZPZ games of varying quality or super EZ PZ shovelware. It'd be one thing if those titles had some trophies and a few hundred trophy points overall. Many of them have platinums, which Sony is ok with because it's profitable.
2. Xbox Achievements: These invented achievement systems, and the varying number of points has its own appeal. It's also nice that you can earn these on PC for select titles. Native PC versions of Sony published games don't support trophies, for instance.
3. Steam Achievements: I've never liked this, though I guess they're better than nothing. Trophies and Xbox Achievements are already pretty worthless to most but at least they are worth certain points and increase your level.
I've never used Retro Achievements.
I hope Nintendo finally adds an achievement system (maybe with Mario Kart iconography like Mushrooms, Flowers, and Stars) on NG/Switch 2. Virtually every other marketplace/console has, and Nintendo is the holdout. If you don't like them, just disable notifications and such like you can on anything else.

Trophies, they just feel really nice.
Getting a platinum trophy never gets old.
Unless you are playing those shitty 10 minute games only made to get trophies.

They are all good. I don't have any preference over Xbox Achievements or Trophies. Steam Achievements could do with having a score/tier and a better way to show them off. Would prefer if the Steam Level on your profile was based on achievements and not trading cards.

But I find all forms of achievements very useful as it's basically like a gaming diary that is automatically recorded. I go back and easily look at what I played, when I started it, when I finished it etc because it's all recorded in Achievements forever. If it wasn't for achievements I wouldn't remember half the games I played last year, which is why I probably can't remember half the games I played on the Switch.

Don't even need achievements really Nintendo. Just a badge you unlock on your Nintendo Account when you complete a game for the first time. That would satisfy me. One "achievement" per game, easy.

I don't particularly care, but Steam achievements are pretty nice. Them not having any value attached to them means the number of them in any game isn't really limited, so there can be as many or few achievements as is fitting.

This.

Only 1 choice there is and that is trophy.
While the other is just numbers lol

Achievements. In the same way I like Coke better than Pepsi.