This is a combination of 2 Issues that essentially renders shipyards unusable.
Problem 1: Shipyards not Competing tasks.
When giving a shipyard an order, it will hang up around 75%, similar to a bug reported here, but was "supposedly" fixed: https://phab.starma.de/T752
Leaving the sector and returning sometimes allows it to complete the task (at least the current job in the shipyard will say "Waiting For Order"), but even if it does, it's not functioning properly. For example, telling it to load a design, leaving and returning to make it complete, then trying to edit the design will give an error in the top right saying "Can't edit design while Waiting For Order"
Problem 2: Not Properly Recognizing Docked Entities Split to T2411
When docking a ship to the anchor, then going to the shipyard, technically it recognizes something is there, as it says "Real: <ship name>" while not allowing any orders relevant to that docked ship such as repairing or deconstructing. Only orders like create new design or load design are allowed. We've experienced this with ships we've built that have been damaged as well as pirates ships that we've captured to salvage. Leaving and returning to the sector does not fix it.