I'm not sure when exactly it happened but the sz-ligature (ß) has been treated as a single character for decades if not centuries. However it's not always behaving like one.
For uppercase, instead of using the actual uppercase sz-ligature glyph (ẞ, see https://en.wikipedia.org/wiki/Capital_%E1%BA%9E) Germans will actually replace the "ligature" with SS. You could argue that this makes it an obvious ligature (although it would then be mislabeled as it's obviously an ss-ligature, not an sz-ligature) but Germans also write out umlauts as the base letter followed by e (ü -> ue, ä -> ae, ö -> oe) and these wouldn't normally be considered ligatures. The sz-ligature is also often called "sharp s", further showing it's not really thought of as a ligature.
According to Wikipedia the ß has been around since as early as the fourteenth century but its history seems a bit nebulous. I would think that it just used to be a common ligature before orthography became standardized in German and later became a character of its own when orthography became more standardized. Modern German orthography actually has a strict rule for picking between ß and ss: short vowels are followed by ss and long vowels are followed by ß (eliminating the ambiguity between Masse (mass) and Maße (metrics/dimensions)).
But another thing ß and the umlauts have in common is that they're not considered part of the alphabet. Children learn the same "Latin" alphabet as in English (except for the names of the letters, obviously) and if anything, Ä, Ö and Ü are added to the end on charts (but not typically in mnemonic songs or rhymes). Incidentally, ß is rarely added in those cases at all (likely because it has no accepted uppercase variant and only occurs in the middle or end of words, unlike A-Z and the umlauts).
And don't get me started on sorting. There are actually two legitimate ways to sort words in German: either you treat umlauts as vowel+e and ß as ss, OR you list ä after a, ö after o and so on. I think the former has become the norm for most intents and purposes but the other one can still be found in various places (including, I think, print phone books).
So, yes it's confusing, but German "Eszett" is not just a ligature and German umlauts are not just vowels with diaereses (unlike e.g. ë in names like "Zoë" where the diaeresis just indicates that the "oe" should be read as two separate vowels). Nobody really knows where this entire mess started but at least we generally have come up with consistent rationalisations for how we use them.
For uppercase, instead of using the actual uppercase sz-ligature glyph (ẞ, see https://en.wikipedia.org/wiki/Capital_%E1%BA%9E) Germans will actually replace the "ligature" with SS. You could argue that this makes it an obvious ligature (although it would then be mislabeled as it's obviously an ss-ligature, not an sz-ligature) but Germans also write out umlauts as the base letter followed by e (ü -> ue, ä -> ae, ö -> oe) and these wouldn't normally be considered ligatures. The sz-ligature is also often called "sharp s", further showing it's not really thought of as a ligature.
According to Wikipedia the ß has been around since as early as the fourteenth century but its history seems a bit nebulous. I would think that it just used to be a common ligature before orthography became standardized in German and later became a character of its own when orthography became more standardized. Modern German orthography actually has a strict rule for picking between ß and ss: short vowels are followed by ss and long vowels are followed by ß (eliminating the ambiguity between Masse (mass) and Maße (metrics/dimensions)).
But another thing ß and the umlauts have in common is that they're not considered part of the alphabet. Children learn the same "Latin" alphabet as in English (except for the names of the letters, obviously) and if anything, Ä, Ö and Ü are added to the end on charts (but not typically in mnemonic songs or rhymes). Incidentally, ß is rarely added in those cases at all (likely because it has no accepted uppercase variant and only occurs in the middle or end of words, unlike A-Z and the umlauts).
And don't get me started on sorting. There are actually two legitimate ways to sort words in German: either you treat umlauts as vowel+e and ß as ss, OR you list ä after a, ö after o and so on. I think the former has become the norm for most intents and purposes but the other one can still be found in various places (including, I think, print phone books).
So, yes it's confusing, but German "Eszett" is not just a ligature and German umlauts are not just vowels with diaereses (unlike e.g. ë in names like "Zoë" where the diaeresis just indicates that the "oe" should be read as two separate vowels). Nobody really knows where this entire mess started but at least we generally have come up with consistent rationalisations for how we use them.