In fact, 同じく does come from 同じ. The key here is to look at the
historical form of 同じ: it was originally a regular adjective,
following the pattern of the shiku-type adjectives (シク活用形容詞).
If it followed the normal development of shiku-type adjectives,
it would have become 同じい, which apparently does occur, if
rarely.  There are some other -ku adverbs whose adjectivial
counterparts are (essentially) extinct in modern Japanese. For
example, 如く (as in 〜が如く) was originally derived from the
adjective 如し, and similarly べく (as in なるべく) from 可し. (The latter
is not a perhaps good example, as some of its other forms
survive, e.g. べき, べからず etc.) There is a (probably) unrelated
morpheme -(a)ku which nominalises verbs and adjectives. Examples
include 曰く←言う and 思惑←思う. It is unproductive in modern Japanese.