Do I need to register in multiple countries for multi-national startup?

I want to launch a multi-national startup. Do I need to reach every target country for legal formalities? Or, is there an international body which could take care of that?

It's wildly dependent on what you plan to do. If this involves doing activities which must be sanctioned/inspected/certified/approved like health, food, finances, military or something else then it may be required for you to get an official presence in some countries. If it's something "neutral" like this Q&A site, then you don't need to register anywhere.

And no, there is no international body which will register your business in multiple countries. But there are sure private companies which help registering companies abroad and provide consulting services. But this gonna cost.