So I'm gonna throw my .02 cents out; but what the absolute fuck is this train-wreck of a travesty of writing? Why is the MC acting like his job loss is 'oh no just a slap on the wrist' and even MORE to the point; his 'daughter' whom just happens to be sitting at the beach, no papers, no evidence to back anything up aside from "oh hi you're my dad, take me with you" "okay." has me tilting my head like 'huh?'
Not unfair points, but I think they're addressable.
Job is simple. I think (though narration/story/dialog/exposition needs to make it clearer) that MC had a decent amount of money. Short term, especially, losing the job doesn't matter, because MC can afford things just fine.
Also, as the firing was unjust, one thing conveyed in the story was that the company could have a lawsuit on their hands, and one that is monstrously lopsided against them. Planning for the future, MC could be banking on that.
He also has a skillset that means finding another job won't be difficult.
So, if you have plenty of money to live on comfortably for a time, skills to find another job when you want, and legal action you could be a part of in your likely near future, there is no need to let yourself be overly stressed by events.
That is, quite admittedly, me reading between the lines, but not such a stretch that it pulled me from the game at all.
The second is easier. He met her at the day, time, and place he annually looked for her mother. She couldn't have known to look for him there and then without knowing who he was, and what he was looking for there. That means, even if not MC's daughter, she has some connection to Emily.
Add to that her age seems to line up with what would be expected, and MC's flashback in the beginning of the game to give us (admittedly meta) reason to believe she was pregnant, and she has another point in her favor.
She also looks a great deal like Emily, from what we could see and what MC says.
Lastly, when he took her to the doctor appointment, a DNA test was done, which means when MC gets the results, he'll have confirmation one way of the other. Which means even if we feel he believed too easily, we'll be given something concrete in game that affirms things.
Oh, almost forgot. Charlie, who seems to either be a PI or a cop, is also looking into this.
So, while MC may seem to accept it too easily, he is also looking for absolute proof. We, as players, aren't being asked to merely hand-wave it all away.