From Encyclopedia Briannica:
New words come into English in many ways (borrowing, backformation, verbing nouns, acronyms, etc), but the most visible word formations in Modern English tend to be blends. Blends (or portmanteaus) are created when at least two words are shoved together physically and phonetically to form a new word. Smog, frenemy, bloggorhea, hacktivist, cronut, phablet, sext, guyliner; they stick out as neologisms, and being visible means they endure a lot of public scrutiny.
So hatarchy qualifies as a blend, which is also a portmanteau, because hat and anarchy have been shoved together and the an- has fallen off anarchy.
Or is it a clipped compound? Encyclopedia Briannica continues:
In her [2014 thesis on English blends] conclusions, [Natalia] Beliaeva makes a smart distinction between clipping compounds and blends, determining that there are different motivations and methods that lead to their shortening and grouping in certain ways.
For her, clipping compounds come from existing phrases such as “National Biscuit Company” and “optical to tactile converter” which are then reduced down to their initial sounds to create Nabisco and Optacon, respectively. Other clipping compounds include NaNoWriMo, Biz Cas Fri, ampersand, Filoli, SoLoMo, romzomcom, sudoopoo, Tribeca, and Dowistrepla. There are two-part clipping compounds too, including retcon, pro-am, sci-fi, sitcom, Pokémon, and MoCap. In clipping compounds, the words are represented by their first segments (similar to acronyms, but slightly longer).
Blends, on the other hand, do not come from pre-existing phrases. Their concepts are brought together and shortened into one word simultaneously. All of the examples from the first paragraph are blends of this kind. These seem more productive than clipping, and usually include the first part of one word, and the last part of the other, resulting in a somewhat natural-sounding word.
No, hat anarchy was not a pre-existing phrase before hatbash, and is thus a two-part blend and not a clipped compound.
So is there a difference between a blend and a portmanteau?
Some do make a finer distinction:
Linguists have also taken up the word and created the technical linguistics term "portmanteau morpheme" to describe morphemes that fuse two or more grammatical categories. Linguists reserve "portmanteau" for this usage only, ignoring Carroll's original intent, and refer to portmanteau words by the less interesting term "blends."
But on the whole I'd say they're generally interchangeable.