Did the books ever clarify if he was walking around in person or as some sort of projection while safe in Orthanc? I can’t remember. Guess it’s time for a reread!
I assumed it was foreshadowing for the Gandalf return
I remember Gandalf saying explicitly that it was not him. We know it was Saruman, just can’t remember if he was in person or some sort of projection.

This Redditor is pretty sure it’s Saruman.

Last time I read the book, I had the same feeling.

So we don’t know if Saruman was there in person or some sort of wizardry type projection of himself. Cool, thanks for the link!
I see why he was the head of his Order.