Why should like types of goods only be exchanged, that is, gold for gold, wheat for wheat, etc.

If not done in this manner, how does it fall into Riba?

In the past, people used to engage in barter trade where sometimes they would exchange dates of a lower quality for dates of a higher quality whereby one would give more of the dates of the lesser quality and get a lesser quantity in return. There could be various  reasons for this; maybe  one  would  have  wanted dates or silver or gold  of  a  certain quality for resale or personal use and had  no money to acquire that so they would resort to barter trade. In the past, money was there, but obviously it was not universally available like today and barter trade played a role. In the present day age, barter trade occurs in some countries and some countries which had economic sanctions imposed upon them had to survive through barter trade. Therefore these principles can still be applicable in today's world.

And Allah knows best.