Is the user interface design of ATC equipment generic across countries?
Most ATC software systems are developed by a few large companies. Companies that produce such software include Lockheed Martin, Thales, Raytheon, Indra, Comsoft, Adacel (and maybe a few others). It's not a big market since there are only a limited number of customers (countries) and safety critical software is very expensive to develop.

One interesting offering is the Albatross open source ATC software. This isn't a complete system (yet?) but some parts are available now. I'm not sure whether this software is actually used in operations anywhere at this point.

Most ATC providers (which are usually run or owned by their government, though not always) do not do their own software development. They buy a system from one of the established vendors, who may do some amount of customisation work ranging from none to extensive, perhaps with a support contract for future fixes or enhancements. There are a few exceptions though, for example Airways New Zealand uses software originally developed by a large vendor but also has an in-house software development group who maintains, enhances, and extends the existing software.

Each system has the same general operating concept - an ASD (Air Situation Display) which shows the locations of aircraft, plus auxiliary windows that may contain all kinds of other things used for communication and coordination. Like any sophisticated computer software, the general idea of two different systems might appear the same, but be completely different to actually operate (think of Photoshop vs. Gimp, or Microsoft Word vs. Apple Pages). 

A qualified controller from the US would not be expected to be able to operate in Australia straight away. Not only will the software be different (different mouse clicks, or different key commands), but also the rules of air traffic control can be different between countries. Furthermore, a controller must become familiar with the local airspace and operating procedures.

Having said that, the general skills of air traffic controllers are portable between countries. On moving from one country to another, a controller would be expected to go through a certain amount of training in their new country, which could range from brief familiarisation to more rigorous retraining.