The phrase 'like many another' is perfectly legitimate, though possibly slightly old fashioned (like me, but I've used the phrase on many another occasion, though not in EL&U answers as it happens). The comparison is with a singular object, which can be stated explicitly or omitted (implied). The word 'child' (or 'student') is optional in the following examples; the sentences are both grammatically correct.
Like many another in his class, John is thirteen years old.
Like many another child in his class, John is thirteen years old.
The phrase 'like many other' requires an object, which must be plural and explicit. Omitting 'children' leaves a non-grammatical sentence.
Like many other children in his class, John is thirteen years old.
The phrase 'like many others' uses an implicit plural object (which must be omitted). Although the comparison is with the other children, you cannot include 'children' in the comparison while you keep 'others'.
Like many others in his class, John is thirteen years old.
You could also write any of the following, each of which uses only one of 'many' or 'other':
Like many in his class, John is thirteen years old.
Like many children in his class, John is thirteen years old.
Like others in his class, John is thirteen years old.
Like other children in his class, John is thirteen years old.
Writing the following has a slightly different connotation; it implies that all the other children in the class are thirteen, which could happen, but is fairly improbable.
Like the other children in his class, John is thirteen years old.
like many other_s_
unless there's another word in the phrase. (So, "like many other_ students_ in his class", but "like many other_s_ in his class".)