Why is sourdough called 'sourdough'? Is it because it is sour to the taste, or for a different reason?
Yes, a well cultured sour dough is actually sour.
The bacteria in the culture, lactobacilli, emit lactic acid as a waste product of their metabolism, making the dough acidic and sour.
Just to clarify (and build on a previous comment): "sourdough bread" generally has anything from a subtle hint of sourness to a strong sour tang.
However, the word "sourdough" is also used in bread-making to refer to natural yeast cultures in general, which can be used to produce many types of bread, including those which are not sour at all. As Optionparty mentioned in a previous comment, frequent feeding will allow yeast to grow steadily, while not allowing the bacteria to produce the acid waste.
Traditional French bakers, for example, may use natural yeast to produce non-sour (or even sweet) breads by following a 3-stage or 4-stage process of building up from starter to final dough. Each stage consists of taking the previous stage and often tripling or quadrupling in size. When this is done every few hours, the yeast grow continuously, while the lactic acid-producing bacteria never get a chance to take over. The final result can be a bread with no discernible sourness, though usually the flavor is stronger and more complex than bread produced with commercial yeast due to longer fermentation and a variety of microorganisms.
In any case, in modern American usage, "sourdough" can both refer to a specific type of bread that generally tastes at least somewhat sour or a process of using natural yeast which may or may not be used to produce sour bread. This latter English usage is extremely old: one can find the word sourdough even in medieval Bible translations used as a generic word for "leavening" in bread.