Does replacing an iPhone/iPad battery reset the battery stats? I have purchased a refurbished iPad. It is from a Apple-certified repair shop, but I understand this is not the same as Apple refurbished.

On a MacBook with an internal battery, I can see in System Information that the battery still has its own firmware and serial number. Is it similar for iOS devices? Does the device know if the battery has been replaced and will this reset battery stats such as loadcycles?

The battery appears to have ~200 loadcycles and only 80% capacity, indicating it is not in good condition. 

However, I have noticed that the charge level regularly goes above 100% when finishing the charge, which might make sense if the charging firmware was "surprised" by a new battery.

My Google-fu was unable to find any battery benchmarks for the iPad 4. (Retina iPad 2012), which I might use to compare the battery's actual performance. I get about 15 hours of screen on time, which seems good.
The battery contains the "battery info", not the logic board. So if you put a new battery, what the logic board reads is based on the current battery only. 

Capacity and Charge are two different things. You're battery can be at 80% of it's design capacity but will still charge to 100% (of said capacity). It's just that a fully charged battery that is at 80% of Design Capacity will not last as long as a new battery that is at 100% Design Capacity. 

As far as your battery, it is at the lower limit. 80% of Design Capacity is okay but as you use the battery and as it ages, it will continue to degrade. The difference between an iPad and an iPhone is that we typically don't use the iPad all day so even if the charge is low a few days later, we just charge it when were not using it. On an iPhone, it is constantly in use (various radios) so we quickly notice reduce battery lifespan. 

Test your battery over time and when it gets below 70%, you may consider having it replaced but it will really depend on how you use your iPad on a daily basis more than anything else.