Well, technically Ashikaga Yoshiaki, the last Ashikaga shogun, gave up the position in 1588, although Yoshiaki escaped from Kyoto in 1573 after being defeated by Oda Nobunaga. In exile, Yoshiaki apparently - unsuccessfully - attempted to garner support by various daimyo, such as the Môri and Uesugi, to recover Ashikaga rule, until he eventually returned to Kyoto in 1588 and officially gave up the post of shogun, becoming a lay monk.

In other words: the Ashikaga shogunate ceased to exist, and the Tokugawa shogunate is a polity entirely distinct from it, although it inherits characteristics of its predecessors (in the same way, the Ashikaga shogunate is a distinct polity from the preceding Kamakura shogunate). Shogun is, in effect, a title the court may give to whomever they conceive being the leader of all warriors, a tradition which harkens back to the early days of the Kamakura shogunate (which, in turn, means that someone who wants to be seen as the supreme leader of all warriors would likely be inclined to want the title to confirm his supremacy), and which came to be transmitted hereditarily in the Ashikaga and Tokugawa lines.

There was no shogun in the time between, and there also is not required any shogun to be, since technically, the shogunate governments are subordinate elements to the imperial court, which always nominally constituted the framework in which they operated.

That the Muromachi and Tokugawa shogunates effectively came to be responsible for virtually all political decisions, and the court more or less only legitimated their existence, is an entirely different issue.

But I'm quite sure u/ParallelPain might have something more to say.