Anglish is neither invented language nor relex nor just a casual / funny "style".
It is, more than anything else, a register. The Cambridge Dictionary says of register: We use the term ‘register’ to refer to particular varieties or styles of speaking and writing. Registers vary because the language is used for different purposes, in different contexts and for different audiences. For example, there is a legal register, a register of advertising, registers of banking and a register of weather forecasting. We commonly recognise registers because of their specialised vocabulary but also because of particular uses of grammar.
Much like how we have registers for differing social contexts ("street talk" or "formal gatherings" or "liturgical worship"), Anglish is simply another register. In this case, I'd argue that the social context is one of distinct national identity vs worldwide loss of identity. Anglish is clearly a distinct variety or style of English --- it is English pure and simple! --- and it is used for the particular purpose of distinguishing English English (Our English) from World English (Everyone's English).
The Anglish Moot goes into some detail here. The Anglish/New English project is intended as a means of recovering the Englishness of English and of restoring ownership of the language to the English people.
In linguistics, a register is simply a "variety of a language used for a particular purpose or in a particular social setting." In the case of Anglish, the purpose is as stated: to communicate using a restored & nativising form of English.
As an artifact, it can't really be called an invented language, because it isn't really a thing "invented". And certainly not in the usually understood sense of the concept of an invented language. For it is already English.
It's not a relex in the conlinguistic sense because it is not a matter of "making up new words" and replacing the English words one for one. (In the linguistic sense, perhaps.) Anglish is a matter, simply, of bringing actual English words, older words, dialect words, disused words to the forefront along with some loan borrowing and calquing from other Germanic languages.
My vote is still for Anglish to be classified a Register of English.