Bonding metal pieces using electroplating - what's the process called?
There's a 1974 Sandia labs paper on "Joining by Electroplating", abstract and citation available here.  They recommend tapering the joint and plating a triangular void closed, which brings up the important point that you need to let fresh electrolyte in to reach the joint as it builds up material.  For example, trying to electroplate two flat disks face-to-face would likely result in most of the deposit forming a ring around the disk edges, sealing a layer of corrosive electrolyte between the disks rather than securely welding them together.

The terminology for this sort of technology has rather permeable boundaries, but includes:


Electroplating, normally aiming to deposit a mostly uniform surface coating over an existing substrate.  Common in a variety of consumer goods, often for corrosion protection.
Electroforming, depositing material over a removable mandrel.  Common in micromanufacturing.
Electrowinning, extracting or purifying material.  Common in mining.