Literally, speaking no.
Personally speaking; I'm no parent, and while it's true Fortnite isn't blood-soaked or anything, I dunno if I'd give it to a six year old regardless. It's still got fairly realistic weapons and it might give them some bad habits once the salt sets in (though I'd say honestly, supervision could go some of the way to avoiding that, at least). If you do let them play, just make sure the mic is off (though I'm not sure if Fortnite even supports mic chat for people that aren't on your team; I just switch it off and use Discord anyway)
If it were me, I'd just keep them on stuff more their intended age group for now; nothing wrong with Mario and Sonic and what have you. But on the other hand, it's not like you're giving them Call of Duty or something, so I suppose it could be a lot worse.
That's not 'literally speaking'. That doesn't mean that the game isn't designed for kids, it means an outside ratings board have deemed it to have a certain age rating based on fairly arbitrary determined values that may or may not align with your personal beliefs and parenting style.
To the OP, I think the best approach would be to ask what your concerns are if they played a game like this? In the OP you wrote...
I have second thoughts now.... should a 6 year old really be allowed to play such a game....what do you guys think?
But I have to ask, what are you concerns here? The literature linking violent video game play with violent behaviour is extremely weak, there's plenty of literature that indicates that within moderation, video games have cognitive benefits. Do you think shooting other players is too violent for a child? It's pretty cartoony violence. While the game does feature real-ish looking guns, they're not real guns.
Ultimately I would ask yourself what your concern is here in terms the potential impact that this could have on your child, and look into whether that concern is well founded. As I said, the literature on video game aggression is very weak. Personally I would be most worried about exposing my child to manipulative monetisation systems at a young age, where their impulse control is quite limited. But that's not what most people are concerned about when they think about age ratings, and many games for children have pretty gross monetisation (e.g. Mario Kart on mobile).
The communications side of thing is a concern too, but if you're only playing in a party together I think that side of the game would be fine. Perhaps there's a way to enable streamer mode so that you can't see the usernames of other players too (you can do this in Apex Legends).
It might also be that, living in the UK I have a softer view on the 'shooting' side of the game. Firearms aren't accessible here, shooting isn't something you can do anywhere other than in a video game. It's a fantasy activity that holds no real-world counterpart. So exposing a child to a game with guns may depend on your environment (e.g. if I kept a gun, then I would probably avoid exposing my children to games where guns were used like toys).
