I have heard of small "easter eggs", such as the priest in Bloodborne saying "Umbasa". There's not much online - anyone know for sure?
Hidetaka Miyazaki has stated several times that Bloodborne is a new game and has no link to either the Dark Souls series or the Demon's Souls series. Series which are not connected either (see this question).
This was never considered as a Demon’s Souls II (source)
The easter eggs have to be considered as "fan service".
An interesting analogy I found on GameFaqs, illustrating the fact that although two games might look a lot alike, they are not necessarily related : "Are Grand Theft Auto and Red Dead Redemption part of the same series?"
Yes, absolutely. You even face enemies from Bloodborne in Dark Souls 3 (flaming versions of sulyvahn's beast are encountered frequently in the chalice dungeons and even the main game for example), and you can find armor sets from Dark Souls in Bloodborne (on dead knights in the dungeons). You can also see the cultural transition happening, and find other hints as well. That said, I don't think they will ever "officially" say so... Just throw it in your face over and over again.