Hi and welcome to the site. My name is Dr. C. and I'll be providing medical information for you today.
I agree that the dating doesn't seem quite right.
The error on an ultrasound done at 13 weeks is about 7 days either direction. That would put the latest day of conception at about December 29th.
It would be better if there was a second ultrasound done EARLIER. The later an ultrasound is done, the less accurate they become. Ultrasounds done at between 6-12 weeks of gestational age have an error of about 4 days in either direction. This ultrasound is at 13 weeks gestational age so the error is slightly wider. Still, I agree that this doesn't add up.
Please REPLY if I can clarify anything else.