Plastic? In part (I think).
Manufactured? Yes.
(The "I think" is not any kind of trick answer; it simply reflects that I am having to take an educated guess at one aspect of the answer.)

Two words? No. I'll tell you for free (because I think it will be no help at all :-) ) that I vacillated between two different ways of expressing the thing on the card, one of which is six words long and does not begin with P, and on of which is shorter and does. I went for the latter.

It is three words including the definite article.

The deer don't stop here? No. *a scrap of applause that quickly dies away, followed by laughter, scattered applause, and much chattering*

Feeling a bit thick. Shut up at the back there.

Abstract and Mineral/Vegetable descriptions of the same thing? Yes. (Good question.)

Figurative? No. *more scattered applause, though*

Game related? No. *amused discussion in the audience*

Part of an animal? No.

Culturally and geographically specific? Yes. I would associate it primarily (and perhaps exclusively) with one country.

UK-based? Yes.

Edible preparation? No.

This one's a bugger, innit?

Human? No.

Animal = one particular species? Yes.

Emblematic? No. *some audience laughter*
Pre 1500? No.

A summary: This is ABSTRACT and MINERAL/VEGETABLE, with ANIMAL and VEGETABLE connections. The answer is three words, including a definite article, and begins with P immediately followed by a vowel. The abstract and mineral/vegetable are different descriptions of the same thing. The mineral/vegetable description is in part plastic (I think) and is manufactured. The abstract meaning is not figurative. The animal connection refers to one particular non-human species, and is not emblematic. The vegetable connection is connected to food but neither it, nor the overall answer, is a dish.

The answer is primarily or perhaps exclusively associated with the UK and dates from sometime after 1500. It is not art (except under a very broad definition), nor a method of cooking. It is not environmental, nor is it game-related.

Three questions provoked odd reactions from the audience, viz: "Passing the buck?", "Is it game-related?", and "Is the abstract meaning figurative? (E.g. the cat's whiskers)". The suggestions that it was art and that it was emblematic also provoked amusement.

Pink Panther? No *considerable audience applause, nonetheless*

Going from the particular to the general.

Fictional beast? Yes. *applause*

Once a novelty? No.

Twentieth Century? Yes.