In my experience, the two are pretty much exactly the same and can be used interchangeably. They’re both just filler words to give the speaker an extra second to think about what he/she is going to say.

"I was like, 10 miles from home when…"

"J’étais genre, à 10 kilomètres de chez moi lorsque…"

They mean the same thing, the only difference is on the language level: genre in this situation, e.g.

Il a genre trois ans
He’s like, three years old

is in French pretty informal (maybe in English too, I don’t know)

This structure doesn’t necessarily introduce a citation, but any form of complement (adjective, proposition, noun, and any citation would be reported speech); it mostly adds an overtone of vagueness rather than exact information.

En français moyen :

Il était du style : “…”

Il était du genre : “…”

Plus construit:

Il était du style à dire/affirmer/… : “…”

Il était du genre à dire/affirmer/… : “…”

Langue parlée:

Il était genre/style: “…”

To be like is an informal way of saying to utter/to say (AHDotEL). The word genre (2) is not part of any idiom using the verb être which would yield a similar meaning in French. Être du genre humain for instance, would mean to belong to. The Académie française discusses it indirectly and labels this sort of usage as an “emploi fautif” (incorrect use) : « Il m’a répondu genre j’en sais rien, phrase d’où toute syntaxe a disparu, pourrait se dire Il m’a répondu à peu près, approximativement, en gros qu’il n’en savait rien. ». One can only speculate as to what the Académie would say about the syntax here seeing there is not even the verb répondre/dire in the sentence you provided; worst, there is obfuscation from a verb (être) which is not even part of an idiom here and which doesn’t mean to say/utter in any way, shape or form.

Ways to express “il était genre“+quote which are more consistent with the French language :

Il a dit un truc du/dans le genre : « [citation] ».
Il a dit un truc comme : « [citation] ».
Il a dit un truc similaire à : « [citation] ».

The difference is that this is correct, if informal, English, whereas it is most likely incorrect usage in French as the syntax is off.