Translation:It is a cat; it does not have a family name.
To express that sort of 'this' you would normally need to use эта instead of это, because it needs to agree with the subject. I see you study French, so think of it like the difference between 'C'est un chat' (это кошка/кот) and 'Ce chat...' (эта кошка/этот кот...). Bear in mind that if you wanted to say, 'This cat has no last name', the French 'сe chat' would work fine as the possessing subject of the sentence ('Ce chat n'a pas un/de nom de famille'), but you wouldn't be able to do quite the same thing with the possessive sentence in Russian, which needs у + genitive case, as well as the genitive ending on фами́лия because of the negating нет. If you wanted to say, 'This cat has no last name', you therefore would need to use, 'У этой кошки нет фамилии'.
https://en.m.wikipedia.org/wiki/English_As_She_Is_Spoke Sounds a bit like this.
The noun is фамилия. Its declension pattern is the same as for other -а/-я ending nouns, except it also uses -и where "мама" would use -е.
Other nouns endgin in ия behave the same:
- Мария →нет Марии →книга о Марии
- Италия →из Италии →жить в Италии
- аудитория →нет аудитории →об аудитории
Such nouns use -ю as their Accusative ending, -и as their Genitive/Prepositional/Dative ending and -ей as the Instrumental ending.