Can "in a like manner" and "in like manner" be used interchangeably, or do each of these phrases have a special meaning? (or perhaps one is incorrect)?
I've seen both expressions in the Google dictionary:
Like is used here as an adjective, meaning
similar to, or the same as, something else
With this meaning, you can indeed find both variants (with and without a). However, "in a like manner" is not very frequent. "In like manner" is more idiomatic, as you can see from this Gngram. It also shows that "similarly" and "likewise" are preferred.
"In a like manner" is probably used having in mind the structure "in + a + adjective + manner/way":
I would like to answer in a similar way.
This other phrase, "in similar fashion", behaves erm... in like manner with the phrase "in like manner":
Factors associated with urban place of birth may behave in similar fashion. (Cambridge)
and
Smaller units can also be staggered in a similar fashion, or fanned out in arcs if a curved shape works particularly well in the garden. (Collins)
Note that the variants without "a" (in like manner and in similar fashion) are more formal and they are set phrases. You cannot replace "like" or "similar" with any other adjective ("in different manner/fashion" is incorrect). Whereas in the phrases in a like manner and in a similar fashion, "like" and "similar" can be replaced by any adjective that would make sense in the context.