While we're in the spirit of ackshually, German has more than five vowels - that is why it needs the umlaut to actually represent all those vowels with the five base letters!
Swedish has 18 monophthongs, Standard German has 14, RP English has 13, so although there's some correlation between the number of vowel letters in a language and the amount of vowel sounds, different languages solve the problem differently.
And in the case of English, horribly. English orthography is objectively crap, where vowels represent multiple vowel sounds depending on context and the history of each word to a ridiculous degree.
i ü ï u
e ö ë o
ä a
Most other systems are very haphazard largely because all of this stuff evolved as scribal abbreviations. (So did umlauts, of course; it just happens to be the lucky case where it ended up mostly consistent.)