Great question - what we built isn't a "nanobot" in the Drexlerian sense (a tiny engineered machine assembling things atom by atom). It's a cell, it runs onn molecular machines like ribosomes, membranes, and enzymes. The self-replication and division you're picturing comes from that biological machinery copying and dividing, not from a mechanical device building a second copy of itself.
So on your specific question: division is very much a core goal of the synthetic cell field, and getting a built-from-defined-parts system to grow and divide reliably on its own is one of the big open problems ahead of us. What SpudCell demonstrates is assembling something cell-like from well-defined components.
Where this does connect to the "programmable matter" dream is that if you can engineer a cell from the ground up, you can in principle program what it makes and does, using biology's own manufacturing stack rather than trying to invent a mechanical one from scratch. That's a slower and messier path than the sci-fi version, but it's the one that actually runs on physics we think we understand.. hope it makes sense!