"We did not finish eating this Peking duck."
It is a common quantifier for Peking duck, e.g. 一只鸡、一盘面、一碗饭、伊梨苹果 and so on, different quantifiers for different nouns just as it is in English. It is used much more often than it is in English, however. This can be seen in the English from Chinese course, e.g. 一名老师 translated a teacher. There's no 'a' in Chinese so this is how it would generally be translated, 'the' is slightly more challenging since the course uses 这些, which sounds more like 'these' to me, although it is the best option I can think of right now.
The speaker probably came with a group of people so they ordered a whole duck but found they couldn't finish it.
Only a small percentage of English words need quantifiers and something like Peking duck would not be one of them. Probably most English speakers who haven't studied an Asian language would not even know that English has them.
So saying "just as it is in English" could confuse some people.
There lies the answer, a group of people can eat a whole duck or chicken and that is what usually happens.
The chickens or ducks I usually see, not Peking ones, are usually roasted, or however they are cooked, whole and then hung on a horizontal petal pole. This is displayed behind transparent glass or plastic in the stall or restaurant. Usually, you can choose which parts you want, or if you are in a bigger group, especially one with the grandparents and all the other family members, you can order half a chicken or a whole chicken, which will be cut up and served in plates. In this way you can order 一只鸡 and have it served in 两大盘 together with whatever condiments are on offer, e.g. 辣椒 (freshly cut green and red chillies)、辣椒酱 (which is usually freshly made almost every morning, if not every morning)、酱油 (soy sauce) and so on. This is just one example of the many ways chicken is sold.
The others are good too. People say 一盘鸡饭、一盘鸡胸肉/腿上肉 or 一盘炒饭/炒面 while 一顿饭, which is the only thing I can think of now, although I'm sure there are many other examples, is often used in the sense of a meal, e.g. 他吃了一顿饭 (literally he 'ate' a meal, since rice and noodles are staples).
Is it common to omit 有 in Chinese?
Other question related to 有:
I've seen some sentences where 有 is being used in front of the verb instead of 了 after the verb (they said it's Taiwanese Mandarin in the comments) to indicate a completed action.
Is that also where 没有 came from?
"有" in the positive sense to indicate a past action is more of a Taiwan phenomenon, but the negative sense is standard everywhere. I don't know how it developed that way.
In my experience, for the negative sense it's more common for "有" to be omitted on the mainland (with the caveat that my mainland experience is mostly in Beijing).
To add to what PeaceJoyPancakes said, 没 has 2 jobs in Chinese (as far as I know.)
1) 有 is always negated by 没.
2) 没 negates past ACTION verbs. This course almost always uses 没+有+action verb. However, in my 4 years of Mandarin in college and 4 months in Beijing, I only ever encountered 没+action verb.
没 is not used with stative verbs, except for 有 which is special.
If you use 不 with an action verb, you are negating a present or future action, or making a general statement.