Is there such a thing as a pin (or pogo pin) clamp for testing?
Look for "test fixture" for PCBs. There are plenty of ideas- generally customized to the board you're using them with, so some kind of custom machining is required. There are companies that specialize in making test fixtures for professional use so they'll have clamps and so on. A "fixture" in this context (as opposed to a "jig") holds the work (in this case, the DUT- the device under test) in position. Depending on what equipment you have access to you might use different construction techniques- CNC machined acrylic plastic plates, laser cut acrylic, inexpensive CNC-milled commercially made PCBs or 3D printed parts. 

There are generally several layers- see this photo from this web page: 



Plus some kind of clamping system. 'Universal' style clamps exist, but in my experience (mostly Asian manufacturing) most jigs are custom made from standard parts for one specific board design, and look more like this: 





If it looks like there is a lot of work in designing and making the test fixture- there is. The design work in making test fixtures and test programs can easily exceed the work for the board design itself. 

For methods applicable to lower quantities and more hobbyist-y applications, consider the methods Sparkfun has disclosed here- using inexpensive PCBs as parts of fixtures. 



There are also reasonably priced cables made by Tag-Connect that can be held in place by hand or snapped in place (the kind of with legs). If you design the PCB according to their guidelines they work decently well.