what does the phrase “as also that of” mean?

Question

Anglicanism was coined to describe the common religious tradition of these churches; ｛as also that of ｝the Scottish Episcopal Church, which, though originating earlier within the Church of Scotland, had come to be recognised as sharing this common identity.

Answer 1

It's an abbreviated reference, to save having to say the same thing twice...

Anglicanism was coined to describe the common religious tradition of these churches; as also [(to describe) the common religious tradition] of the Scottish Episcopal Church

Answer 2

that = relative (anaphoric) pronoun referring back to an earlier noun or noun-phrase

as = in the same way, similarly

also = in addition, as well, too

and also too that [i.e. the common religious tradition] of the Scottish Episcopal Church

Compare:

This "Renaissance" of the Latin language (as also that of ancient Greek) did not start any earlier than the 8th-9th century.

as also the "Renaissance" of ancient Greek...

and the "Renaissance" of ancient Greek as well