I danced from age 3-20. I took ballet and tap when I was 3 and definitely remember ballet being very serious.
OP- to be honest, this teacher doesn't seem out of line for a ballet teacher. If she is anything like the ballet teachers I've encountered over the years, they think they are doing your DD a favor. They are getting her ready for the "real world" of ballet, yes even at the age of 3. Whether or not your daughter becomes a ballerina (heck, she's only 3!) this is how it'll be in the Dancing World.. trust me, I've been there.
OP- if your DD seems very unhappy and doesn't enjoy the class, I'd say pull her out. Some dance studios are more serious than others. It could be that you are at a studio where girls are trained to be prima ballerina's from age 3. If you're looking for something more fun, I'd say move her elsewhere.
EDIT: Just saw your update.. that's great!! Hope it goes well!