#trivial similarities between Bruce and Joker are especially interesting to me

[distort-opia]

Have joker and batman ever agreed with something (not begrudgingly) ? Always been curious about this

They've even teamed up occasionally in the past, but I wouldn't count all of these instances as agreeing on something. It was out of necessity, most of the time. And explicitly getting Batman and Joker agreeing on something is quite rare, but there are things they've seen eye to eye on. I'll try to go through them below, but I'll put the elaboration and the comic references under the cut, since this got long.

Bruce agreed with Joker that he'd been making mistakes as Batman after Alfred's death;

Both Batman and Joker hate The Batman Who Laughs;

Both Batman and Joker agree that the world is meaningless and absurd (but they disagree on how to deal with it);

Both Batman and Joker believe in Joker's insanity as a reason for him not being fully responsible for his own actions;

Both Batman and Joker see their sidekicks as better versions of themselves, not imitations;

Both Batman and Joker oppose Nazis.

1) Bruce agreed with Joker that he'd been making mistakes as Batman, in Batman: Joker War.

This one is probably the most straightforward:

Batman (2016) #94

Batman (2016) #101

2) Both Batman and Joker absolutely loathe The Batman Who Laughs.

Neither wants Bruce to actually become him, they both think TBWL is an abomination-- which results in Joker teaming up with Bruce to help in defeating him two separate times (Dark Nights: Metal #6, The Batman Who Laughs). It's to the extent that Bruce relies on Joker to kill him if he turns into another TBWL (The Batman Who Laughs #4), and to the extent that Joker pretty much forbids Lex Luthor from working with TBWL, wrecking the Legion of Doom when Lex does not keep his word, in Justice League (2018) #13:

3) Both Batman and Joker agree that the world is meaningless and absurd (but they disagree on how to deal with it).

This is perhaps most evident in Batman: The Killing Joke:

It's been alluded that they agree on this in other stories too, like Batman: Ego and Batman (2011) #48. But TKJ remains the most poignant, since the fact they share a loss of meaning due to trauma they couldn't process is at the basis of the story-- and the point of it is that Joker is wrong that the only possible path after experiencing something like it is madness. "Maybe I've been there too," Bruce says. Despite Batman and Joker disagreeing on the how of dealing with it, they definitely see the world in the same way.

In the same vein, there's this moment in Batman (2016) #48 that's nothing if not agreement regarding the construction of their identities as a way to keep themselves alive in the wake of loss:

4) Both Batman and Joker believe in Joker's insanity as a reason for him not being fully responsible for his own actions.

Over time, more than one character has gotten at the truth of Joker's persona being, in many ways, a desperate performance; Jason dug at it in Batman: Under the Red Hood, Jim ponders it in Joker (2021). But Selina spelled it out best in Batman/Catwoman, even though technically this isn't within continuity:

Batman/Catwoman (2021) #9

"You're like him! Just exactly like him! [...] You know wrong from right, and you just chose a side!"

Most of the time, if Bruce and Joker ever agree on something, it's that Joker is crazy. They both need to believe it, for different reasons. Joker chose "madness as the emergency exit", and if he admits that he isn't mad, he lets go of the single coping mechanism he's constructed in order to keep surviving (not to even delve into the fact that this is how he deals with his breaking of the fourth wall). Although this isn't to say Joker isn't mentally ill (because hoo boy, is he)-- however, he isn't irrational. He's not the chaotic and monstrous force of nature he presents as; he's a human being making choices. And also... because he's "crazy", Bruce pulls stunts like saving Joker's life from the electric chair (Joker: Devil's Advocate), saving him from the Spectre's judgment (The Spectre #51) and from the demon Etrigan (Batman #546), saving his life when he gets stabbed despite Jim stating it's a consequence of his own choices (Batman: Cacophony #3)... by arguing Joker is a psychopath and incapable of making choices. In Batman/Catwoman, it's to the point he makes Selina swear she won't kill Joker, with Selina keeping to herself the thought that Joker was sane, indicating exactly just how much Bruce needed to believe it.

I'll make the note that this is of course my personal interpretation, and that there are times Bruce has referred to Joker as not insane. Like in Batman: Endgame, in which he calls Joker "not crazy, just evil". However, that's kind of the point, at least in my understanding... he only does this at the end of the road. And Joker has also called himself "differently sane" or argued against being called insane in actual courts of law, but his reasons for doing so are usually creating more chaos or a wounded ego (not sincerity).

5) Both Batman and Joker see their sidekicks as better versions of themselves, not imitations.

Well, according to Bruce at least:

Punchline: The Gotham Game #2

6) Both Batman and Joker oppose Nazis.

This is a less personal one, but Joker asserts more than once that he's against Nazis and bigotry, in the Batman & Captain America crossover, and in Injustice: Ground Zero.

And there you go, Anon! I tried to limit this to beliefs agreement can be applied to, since going into the emotional similarities Batman and Joker share is something else. Also, if anyone has other cases in mind, feel free to add on!