Is this unusual/illegal for a private duty nursing agency?

Hello! I am a newly-licensed LVN in Southern California, and have started my first job at a small "Home Health" agency as a private duty nurse. Upon being hired at the home heath agency (which I will refer to as agency "A"), I was told by my boss (the owner of Agency "A") that I would have to tell my pts' families that I actually work for a different agency (which I will refer to as Agency "B"). So, basically, Agency "A" issues my paychecks, but my documentation and time cards are on Agency "B" forms, and I am supervised by RN's and case managers in Agency "B."

I don't feel comfortable lying to my patients families, and the subject came up the very first day, so I told them the truth and they reported this to Agency "B," who told Agency "A," and I was then gently reprimanded. This all seems so odd to me, and I don't know any nurses in real life that I can ask for advice on this matter so I am turning you all here!

Is this legal? Is this ethical? It feels very, very wrong to me...