Formal adoption wasn't really legal back then during 19th century England and neither of them truly considered each other brothers. Literally one of the first things Jonathan says after the rugby match is how he can't see Dio as a "real brother" and Dio most definitely didn't consider Jonathan a brother either.
A lot of people ship Dio and Jonathan because of Dio's one-sided obsession with Jonathan as well as Dio's attitude towards Jonathan in the ship where he pretty much talks about how he's incomplete without him and how they're two halves of a whole. It's this idea of a twisted obsession with someone where you don't ever let the other person go and instead literally sew your bodies together for eternity.
When Jonathan dies, Dio is genuinely distraught. I wonder if when he said "you can live forever with erina" but really meant "you can live forever with me, i'd be alone otherwise, you who cried when you struck me down, you who cared for me despite my hatred for you, you who might be the only one who could ever understand me in any way" and I can see why jonathan x dio is so popular lmao.
There's also this commentary by the literal director of the Phantom Blood anime where it seems like he's hardcore shipping Dio and Jonathan lol. He calls them "brothers, rivals and even something like lovers".
There's also this 2007 interview by Araki in Eureka Magazine where he says Dio could've had "lust" (or "desires" depending on the translation) for Jonathan.
It's sometimes argued that the interviewer was a fujoshi and she was pressuring him to say that but reminder that in the same interview, Araki was completely dismissive of anything romantic between Bruno and Abbacchio but the fact that he wasn't with Dio and Jonathan says a lot.
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