It's a little confusing, but if you want to go even further back the series is the spiritual successor to King's field from the PS1/PS2 era, elements of which are in the souls games.

But otherwise you're kind of right. Demon's souls was made with SCE Japan Studio, and it's SONY's IP. Dark Souls is the spiritual successor to Demon's Souls, but From themselves own the IP for it, and worked to it on their own. Dark Souls 2 is similar. Bloodborne is made with SCE Studio Japan again and is funded by SONY. I often get the impression with Bloodborne that from initially wanted to name it something like "Beast Souls" or something (as hinted by the initiallly leaked trailer which stated the games working title was project beast - which similarly for Dark Souls, was called project dark), but they maybe wanted to emphasise the Blood elements more.

An Alternative reason for it being called Bloodborne that i've seen being handed about is that SONY are trying to make their first party releases this generation not tie into previous series so much - for example, Killzone Shadowfall and Infamous second son not being called Killzone 4 and Infamous 3 respectively, in order to not put off new people to the franchises who won't have played the previous games (because lots of people are new to the PS4)

They also couldn't really make an exclusive game called Dark Souls 3 because that's From's IP, and calling it Demon's Souls 2 really wouldn't fit (because Demon's souls were a huge plot device in Demons souls), so they probably just decided on something that SONY could market well and fit with the game thematically, especially as souls aren't really a thing.

It's a bit of a mess of naming, admittedly. Just call it King's Field 8.