Yep, the head is a Gillette Old Type from 1910.
https://www.badgerandblade.com/forum/wiki/US_Gillette_Dating_Information
I’m having a lot of trouble identifying the handle though, those older Gillettes almost always had knurling on the bottom… http://www.mr-razor.com/Rasierer/Old Type/Old Type.htm
The head looks authentic to me. Very common to replace the handle of a 3 piece razor if the handle breaks or to make it look different. Lots of companies were making compatible parts.
But it might be another Gillette handle there just aren’t good pictures of on the Internet, they made a lot.