I rarely do this but you do seem to genuinely be asking, so worth a shot. Short answer: It's up to YOU.
Comics are VAST… there are different writers, different arcs and on top of that, different batman medias. Batfam, in general terms, falls in the "found family" category. The bonds someone makes can run deeper than blood, I think most people can agree with that. But given the vast number of issues they have, a lot of people especially older readers will see them as allies or brothers in arms and not necessarily as brothers (siblings). You can share a father figure (such as Bruce or Alfie) and that doesn't mean you consider every other person - that sees him as father figure - as your brother (sibling). I can tell you I consider my boss and mentor like a father figure, that doesn't mean his wife is my mother or his kids my siblings lol. Think about it yourself: would you?
Have they (Jason, Tim, Dick, Damian) called each other "brother" in comics? short answer: yes. There are at least 3 panels that come to mind in recent comics. The same way that there are panels in which they berate, attack, outright antagonized each other and not in the "fighting with your sibling way" but in the" strangers/acquittances/former allies meeting as enemies" kind of way. There's also panels in which they outright say they would never see them as brothers (such as Tim when Damian showed up, after Damian literally tried to kill him; or when recently Damian was very happy to learn "HE HAD A BROTHER" in reference to Respawn). Very quick rundown of what I know: Dick and Jason barely had much interaction before Jason died - primarily because of Bruce and his really shitty way to show care. He fired Dick from being Robin hurting Dick in the process; he was an adult so he left.
Bruce found Jason almost immediately afterwards and then Jay became Robin. When Dick found out about this he was obviously not happy about it. Jay and Dick had a rocky relationship at first because of the mantle and because of how Bruce handled it. They made peace and Dick accepted Jason as robin but Jason didn't grow up with Dick as an older brother. Dick had already moved to Bludhaven, he didn't live them. Dick was primarily the first Robin and the shoes he had to fill. After Jason died, Dick was obviously crushed and remorseful but again… they didn't grow together so they didn't see each other as brothers. Moreover, when Jason came back as Red Hood, Jason was NOT the "rebel boy"/anti-hero/batfam black sheep he is in current comics. He came back as an straight up antagonist.
With Tim is even worse: Tim only came into the picture AFTER Jason died. Their first interaction was as enemies. Even after the events in titans tower and Battle for the Cowl - where they are enemies - their relationship is extremely rocky. It's only in New52 where DC forces Jason to make a shift and starts slowly trying to integrate him back into the batfam. The only ones that had what could be called a sibling relationship in parallel from being robins were Dick and Tim; and Dick and Damian. Dick was more present for Tim as an older brother while Tim was Robin. When Damian's turn came, Dick essentially took full responsibility of him while Bruce was presumably death (and ditched Tim which brought a another series of issues between them and made Tim stranded). Lastly, the adoption part is also extremely tricky… in older comics Jason and Dick were only wards (which is an entirely different legal figure than adoption). For what I know Dick was only adopted as an adult… and Tim? his parents had been alive even while he was still wearing the robin mantle… he was adopted later but that adoption was erased post flashpoint. I honestly don't even know wtf is going on there currently - and don't personally care. I simply choose to go with the older comics and I leave it at that.
And THEN you have other batman medias outside of comics…
in Arkham games Tim is strictly an ally/partner and nothing indicates he was adopted or that Bruce was even his legal guardian; in Titans TV, Dick and Jason were trained by Bruce but they didn't even grow up together, Jason only became Robin and was living with Bruce AFTER Dick became Nightwing and Dick had very much cut his relationship with Bruce; Jason died and became Red Hood; Tim, wasn't even trained by Bruce, he was trained by Beast Boy lol and he directly joined the Titans as Robin, didn't even have a close relationship with Bruce; Jason and Tim only meet as enemies at first; later they do make peace.
In short, what I'm trying to say is… there are decades of comics for these characters. Bazillion writers and stories. On top of that you have different movies and medias. Can you REALLY blame people for having different interpretations when DC goes back and forth with their own "canon"? Some people read comics 20 years ago and decided to stop; some people decided to hop in 10 years ago and don't look back; others decided to star reading now because the stories seem more friendly and less dark. I think everyone is ok to pick comics at whatever point they prefer; and no one should be forced to know absolutely everything - and that doesn't make them right or wrong. I PERSONALLY interpret the batfamily as a found family with DIFFERENT kind of relationships and dynamics. To me the father figure, albeit shitty, is Bruce, but this is independent from each batfam member and their relationship between one another. I for one, only see Dick and Tim; and Tim & Damian as brothers; the rest of the lot: not necessarily - ESPECIALLY when it comes to Jason. I definitely see Jason and Bruce as father and son; more than anyone else imo. But where it comes from Jason and everyone else, it is a different story (at least to ME): again, just because you share a father figure, it doesn't make every other person your sibling. Family isn't a right and it's a privilege and it has to be earned. Calling someone a "brother" doesn't necessarily make him your sibling. It could easily be brother in arms. A friend or bestie is someone we call brother more often than not. To ME: there being 3 or 5 panels of them referring to each other as brother or sibling doesn't erase the fact that they have spent the majority of their lives being the exact opposite. There needs to be a bond, a relationship, something to make that word mean something. If for some people a word one writer introduced is enough, I think that is totally ok! whatever makes one happy or if that is how some people were introduced to these characters, that's valid! but other people's interpretations are also valid especially in this specific setting where there is so much history and not just a single answer.
I don't know if you will read all this but I do hope, in good faith, that you will at least respect other people's interpretations. We can all share space if only people learn to be respectful.
