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You can go a lot of places from Aberdeen Pier in Hong Kong, though they are mainly outlying islands.
According to the Transport Department in Hong Kong, there are two franchised/licensed ferry line that departs from Aberdeen Promenade that gets you to Lamma Island:
Furthermore, according to Traway Travel Limited, they operate two more routes (in addition to 1.) from Aberdeen as of 2019:
Aberdeen - Stanley / Po Toi Island
Aberdeen - Cheung Chau (Alternate link)
There are also two river crossings (kaito) routes between Aberdeen Promenade and Ap Lei Chau (main street), and a ferry service that gets you to the Jumbo Kingdom, a pair of floating seafood restaurant nearby.
The OP also asked:
Are there ferries from Aberdeen to [...] Mui Wo (or other parts of the Lantau Island), or to Central, or to the Kowloon Peninsula?
As of 2019, no, at least not directly: