Okay I actually think this little scene is hella important. We all know HYDRA did all kinds of experiments on him before Steve found him. Who’s to say they didn’t experiment with wiping memories as well ? They probably already started all kinds of experiments related to the Winter Soldier on him.

See how Bucky’s rambling at the beginning ? It’s like he’s trying really hard to remember who he is. He keeps repeating numbers and his name like it’s something he remembers but he doesn’t know what it means, he doesn’t know it’s him.

Then when Steve shows up he doesn’t recognise him at first. It actually takes him a good couple of seconds before he remembers who Steve is. And with remembering Steve, he also remembers himself because Steve is part of him. I’m going to curl up under the blankets and cry myself to sleep now.

It’s the opposite, but I see where you’re working this through.
In case you don’t know “name, rank, serial number” is the (romanticized) formula that US armed forces use under torture and duress. i.e. you don’t owe your captor anything more than required to identify you to the Red Cross as a prisoner of war. You don’t have to give away your unit, troop movements, equipment, FAMILY names and locations- all of those things become possible tools the enemy can use against individuals and the war effort.
So there’s this whole bravely suffering soldier trope of having repeated name-rank-serial number for so long under torture that when you’re rescued you’re still in that fugue and you’re repeating it on auto-pilot.

Short version, no, he’s not trying to remember who he is: likely that is all he can remember EVEN if it means nothing to him and he doesn’t know why he’s saying it. Torture.

I totally dig the name-rank-serial-number thing, but IIRC in this scene he’s saying it out of order.  He’s fumbling through most of the number; he’s got the name mostly down, but not entirely.  The ‘I can’t remember who I am’ vibe I got was that, of all the things a soldier should remember, this one thing is key.  This should be the mantra that you can say no matter what. (Romanticized, yes.  I’m talking ‘movie universe soldier’ type of stuff).

But even that.  Even that…. Bucky doesn’t have it right.  He can’t quite get it out right, not just because he’s exhausted, but because I think he doesn’t really have the cognitive abilities, right then, to get it right.

So.  Yeah.  Second verse, same as the first, hehe.  We’re both right.