Were There Any Museums In The 16th Century?

One museum which was basically an Art museum was called Uffizi and it was in Florence. It was the first publicly owned museum in Europe. It was established in the 16th century but at that time it was open for visitors only on request. This was one of the first modern museums and later on in 1765 it was officially opened to the entire public.
Some of the other old museums of Europe which were opened in the seventeenth and eighteenth centuries are the Ashmolean Museum of Art and Archaeology, the Museo Sacro, and the British Museum in London

Yep, and even much earlier. Greeks, romans, chinese, egyptians, many cultures had what we would now call museums. And these were established thousands of years ago. The europeans were definitely not the first.