Why does each chasidut have its own distinct clothing?
Google and Wikipedia only list different kinds of clothes, but they don't explain why.
Nothing to do with chassidus but traditions over years from what country and cities their families cam from.
Much of various Jewish clothing is loosely based on what the Jews wore in Babylonia which included a long garment and a round hat.
Of course over time styles have changed and evolved to what they are today.
As far as any connection to chassidim (not chassidus) many simple wear what their Rebbe wears but this is not necessarily always the case as personal preference can vary from individuals as well as personal and family customs.
Because that's what they've always worn.
Most likely, different number of buttons / slits / color of socks / height of hat / etc, started from some trivial, historical detail, which has long since been forgotten - for example, that's the way the local tailor happened to do it, or a specific rebbe just preferred it that way...
And, since "this is the way we've always done it", it has become a big deal, and customary within each sect. And of course, it's forbidden for them to change anything that has always been the custom.
Reminds me of the story of the rebbe and the broom in the corner...
To some degree or other, most Chassidish clothing looks similar (I.e. The bekishe and Kopote look relatively similar, the only difference is the amount of buttons or slits). Different Chassidim come from different parts of Europe where there were different styles. Then as they came to America, each picked up from the local style to some degree or other. Some styles also changed from Europe because of a lack of money of some of the immigrants (The Lubavitcher Rebbe had to reintroduce the Kopote as they were forgotten due to the poverty of the immigrants coming to America) .