I don’t think Vivian stole the embryo…I think Kate just wasn’t able to have the baby, and so Vivian was willing to. And yeah, Philip knows that she’s the one who gave birth to him.
Vivian found out that Kate was going to be implanted with her and Victor’s embryo, so she went to the hospital and switched the petrie dishes so she got Kate’s. Vivian wanted Victor and at the time, Victor and Kate were in love. So Vivian stole the embryo. Kate and Victor were furious when they found out but the only option would have been to force Vivian to have an abortion and then they wouldn’t have their child. Vivian was made to surrender the child when he was born.
I remember when Viv stole "Phillip" from Kate! Those were the good ol’ Days when they had scripts like this…Vivi sure lights up Salem…I think it would be funny to have Carley walking by a graveyard and have Viv step out from behind a tree and go "Boo!"