The history of the Red Cross is not as old as it may seem. It was invented in 1864, during the first Geneva Convention. It was a congress of European countries which wanted to establish common rules for the care of the wounded on the battlefield. Without going into detail, I will say that the countries then agreed that it was a good thing to care for the wounded - their own and others' - and thus medics should be able to do so without being attacked by the enemy. A simple and clear symbol was needed to signify the medics during battle.
As the Convention was held in neutral Switzerland, the delegates suggested using the "reverse" Swiss flag as such a symbol. The red background became white and the white cross in the middle became red. Thus the Red Cross was born.
All the powers recognised it, and began to use it in wars. And all was well for about 14 years, until the Russo-Turkish War. (The one in which the Russian Empire pushed the Ottoman Empire out of the Balkans, thus giving independence to Bulgaria and Romania.)
The Turkish government did not want to use the cross, believing that the morale of Muslim soldiers would suffer if they had to fight against the Russians with such a symbol in their own ranks. They perceived the cross as a Christian symbol that reminded them of the conflicts of the Crusades.
As a result, the Turkish army used the Red Crescent in battle. Communication with Russia had to be established through diplomatic channels, and it was agreed that Turkey would respect the cross on the Russians, who in turn would treat the crescent in the same way.
Here we go! If Muslims can have the crescent, why not us? - Other countries asked. The Persians wanted a Red Sun, the Hindus wanted a wheel (which they have on their flag), and Holland wanted to use the triangle... There were many other suggestions. The symbol approval community understood that they could not go on like this - if every country had its own Red Pretzel, the recognition of basic symbols would fall, and soldiers would be tired of memorizing them all. So the Red Cross and Crescent Association eventually persuaded everyone to adopt either the cross (India, Holland) or the crescent (Persia).