Over text? Where is he 'working away' that you couldn't have facetimed him or even voicecall rather than a text?
You can't judge body language on a text and now he has the heads up to get his story straight.
I would bet you every penny in my savings account that the boy is genuine.
He either is your partners son OR at the very least, has been told by his mum/others that he is.
He's approaching you first because it's bloody scary and huge and you feel a slightly less scary option - like a teenager getting his mate to ask a girl out rather than asking her himself.
And as for not giving you his mum's name, that's perfectly reasonable too if she doesn't know or doesn't want him to contact the father. He might tell you if you promise not to contact his mum without discussing it with him first.
A birth certificate will only prove what the boys name is - you can't put a fathers name on without him being there and IF your partners name were on it then, he's a big fat liar ... but, he's likely not on there, because he either did a runner when told about the pregnancy or maybe he didn't even ever know - that would be the best outcome for you and your relationship.
Questions to ask the boy are - how old are you? That will help solve the mystery. If he's under 16 at the least, I'd tread very carefully. They're brains & emotions aren't developed enough for this kind of thing - encourage him to talk to a safe, trusted adult in his life about this even if that person isn't his Mum.
Ask where he was born, where his Mum and Your partner met.
Ask him if he's tried contacting your partner directly, now or previously - it could be that he has and your partner hasn't seen it (sat in message requests) or has ignored it or even that he's aware of and has blocked the boy already.
Teenage boys with absent Dads is such a minefield (a totally unfair one for the child) and it's why I've tried so hard with my own son who's a secret in his Dad's new life. My son knows that if he really desperately wanted to contact his Dad, that he'd have my support (though I would try and get him to wait until he's st least 18 if not older) not all kids are that lucky.