I use Shimano stuff, I'd recommend that, but any bike specific mineral oil will be bike specific at least, anything else will probably work to a degree but may or may not be too thick or thin or have additives that may not be ideal for the seals or work as well at temperature. You'd have to test your other options to see how they work if no one else has already.
Yep Bike Tart are fine for Shimano stuff. I've also used the fluid by epic bleed solutions, direct from their website. I also often order both from Amazon.
Direct from Madison with a trade account. You were very much not clear in your questioning or I misunderstood as it was muddled up with talk of baby oil and different fluids.
I use Shimano stuff, I'd recommend that, but any bike specific mineral oil will be bike specific at least, anything else will probably work to a degree but may or may not be too thick or thin or have additives that may not be ideal for the seals or work as well at temperature. You'd have to test your other options to see how they work if no one else has already.