https://www.gov.uk/permission-take-child-abroad
There's the official information.
Yes, you need the child's father to give you consent. In the same way that you would need to give consent were he to take the child abroad.
Do not listen to incorrect, and potentially dangerous misinformation.
All you need for the majority of countries is a signed letter with his details saying he has no objection. You can download templates.
Surnames have nothing to do with it, beyond acting as a potential red flag if there were already suspicions raised at API or check in stage.
Think of it as like a customs check. We all know you aren't allowed to bring certain things into a country, some people are stopped and checks made, others not. That some people have never been stopped, doesn't mean it doesn't happen. "Only on Mumsnet" do you see people thinking this.
I live in Italy and come back and forth several times a year. DD is now 20 but I was probably stopped about a dozen times in the years when she was a minor. Never any issue because I had the consent letter. I was questioned more times leaving Italy than entering the UK. (Both of us on British passports)
Before I came to live here, I actually worked in this department and at my interview it was the very first question asked. Why should border control pay special attention to a child travelling with just one adult.
Don't worry about it @Pertue , but do get the letter. And have a great holiday.