The expectation’s not entirely unreasonable, given that the local names of many if not most countries on the Continent do sound at least vaguely like their English exonyms. For instance, there’s:
[ul]
[li]Spain - España [/li][li]Italy - Italia[/li][li]France - (La) France[/li][li]Belgium - Belgique/Belgien[/li][li]Switzerland - Suisse/Svizzera/Schweiz[/li][li]And so on[/li][/ul]
As for the case of Germany in particular, the word deutsch- ultimately traces back to one meaning “of the people”, at least according to Wikipedia[sup]1[/sup]. I have to wonder if this is related to the word deutlich, meaning “clearly”, as in speaking or writing. If so, it would seem that the Germans’ name for themselves, die Deutschen, originally meant “the people who can speak clearly, like us”.
Given that the Germans’ ancestors, in the early Middle Ages, comprised a number of distinct tribes, each with its own homeland, traditions, and rulers, but most of whose dialects were more or less mutually intelligible, it’s not entirely surprising that there are so many names for the country today. The French name Allemagne, for example, comes from the name of one of the tribes, the Alemans. Within Germany, the names of some of the Länder themselves recall the names of the tribes that ruled or lived there, like Hesse, Franconia, and Bavaria to name a few examples.
[sup]1[/sup]From the German language Wikipedia, here.