It's a common enough suffix.
The French -at is cognate with English -ate, from Latin -ātus, which is the Latin perfect passive participle (or from the Latin -ātum, which would be the supine). That also means -at in French has the same root as the verbal past participle -é; but of course the two are used very differently.
The venerable TLFi has a whole entry on -at, which splits the senses up into different headings and some examples of each. Note the English "cognate" may use different suffixes.
I.− Le dér. indique une manière d'être en relation avec le sens de la base qui désigne toujours une personne.
A.− Le dér. désigne une dignité, une fonction et plus vaguement un comportement (permanent ou passager, l'état d'une ou de plusieurs pers.).
E.g. Fr. émirat, palatinat, sultanat, voïvodat - compare English emirate, palatinate, sultanate but voivodeship.
Fr. inspectorat, patriarcat; compare these with English inspectorate but patriarchate vs patriarchy, which bears a split in English along semantic lines.
The "behaviour" category is probably the most likely to have a different English suffix; you already have Fr. anonymat vs Eng. anonymity, and I'd say Fr. assassinat vs Eng. assassination (although false friend alert: it's more likely to be equivalent to Eng. murder/homicide, depending on the specifics) is another good example.
B.− Le dér. est un nom coll. désignant un groupe de pers. exerçant la même fonction ou ayant le même comportement, le même mode de vie.
C.− Le dér. désigne un système, une organ. soc., jur., etc. :
Compare Fr. prolétariat to Eng. proletariat (note the -at rather than -ate, being a 19th century loan from French into English), versus Fr. syndicat vs Eng. cognate syndicate, but by extension trade union.
D.− P. méton. Le dér. désigne le lieu où s'exerce la fonction (la dignité) ou le temps qu'elle dure.
This is an extension of A, more or less, although TLFi do put Fr. juvénat here; although juvenate is attested in English, juniorate seems to be the preferred term in the Roman Catholic church in the Anglosphere.
II.− Le dér. désigne un produit industr. ou le résultat concr. d'une action.
A.− Produit chim., pharm., etc.
There is a plethora of scientific terms here, this being a productive suffix in the scientific world within many European languages. Fr. substrat vs Eng. substrate, Fr. distillat vs Eng. distillate, Fr. filtrat vs Eng. filtrate.
Note though how Fr. hydrate uses -ate instead, with the same spelling as English hydrate; chemical nomenclature for substances in French requires -ate.
B.− Plus gén., résultat d'une action.
Fr. lysat vs Eng. lysate, Fr. infiltrat vs Eng. infiltrate, but only in its noun form.
Then you have lexemes like Fr. crachat, which is pretty transparent to a French speaker, but doesn't have a cognate in English, and would just be a gob of spit in English if being familiar or sputum for a more elevated register. Similarly, Fr. odorat, which would simply be sense of smell in English.