Maybe. But "old man" is slang for father, and Father is a title given to Shaun, specifically, for his contributions with the gen 3 synths. We don't really know much about other directors, but the director when Shaun was taken was probably a woman, given the Director's Recording 108 holotape.
I got the sense Kellogg knew who Shaun was in Fort Hagen when he talked about him being safe and happy at the Institute but a "bit older" than the Sole Survivor might expect *wink wink.* Also, Father is widely known throughout the Institute, even low-on-the-totem-pole synth workers know his story and profess to adore him. It doesn't really make sense to me that Kellogg would be kept in the dark.