Gestational age is very hard to understand. Back in the day when they didn't have accurate ways to measure (or even confirm) women's pregnancies, they measured them from the first day of the woman's last normal period. Because they had no better way to count a pregnancy. But women don't get pregnant on the first day of their normal menstrual cycle, they get pregnant about two weeks after that. But the medical "dating" of the pregnancy starts the first day of her last period.
So. Her doctor does think she got pregnant around May 19, even though you aren't quoting the doctor as saying it. This girl's doctor believes she got pregnant on or around May 19, whether or not he has said that to your face. By stating she's 13 weeks 1 day pregnant, he believes she got pregnant 11 weeks 1 day ago, approximately.
You might want to google this again, "dating a pregnancy" to get maybe a clearer answer, but this is a fact - if she's 13 weeks pregnant, she got pregnant about 11 weeks ago. Silly, I know.
Very early ultrasounds - like at 6 weeks - tend to be very accurate. For a 13 week pregnancy to be six days off isn't that far fetched.
Can you ask this girl if she had sex with some other guy nearer May 19? Do you think you'd get a truthful answer?
Either way, you'll probably need to DNA test the baby once it's born.
Actually, if she was 13 weeks 1 day pregnant on Aug. 5, that means she had sex around May 19 to get pregnant. To determine when she actually got pregnant, you count back only 11 weeks 1 day. The way pregnancy gestational age is calculated you count back to the first day of your last regular period - if she can't remember when that was, the doc measures gestational age and subtracts 2 weeks to get the age of the pregnancy. Sorry, this is incredibly unclear.
But anyway, Since you had sex May 13, and the doc thinks she got pregnant around May 19, you are well "in the window" for it to be yours. 13 week ultrasounds can be off by that much - less than a week.
Do you know whether she had sex with another guy closer to May 19?