celebrating our common values as a people

[latenight]

"It is one of the great patriotic symbols of our country. It is a symbol of the American spirits; it is a symbol of unity; it is a symbol of our values. So for all those reasons it's an appropriate place to celebrate an inauguration that is really built around celebrating our common values as a people." ---- Linda Douglass

Is it common or grammatically correct to use "as a people" here in the sentence above to describe "our common values"? But "values" are different from "a people", right?

[abaka]

"It is one of the great patriotic symbols of our country. It is a symbol of the American spirits; it is a symbol of unity; it is a symbol of our values. So for all those reasons it's an appropriate place to celebrate an inauguration that is really built around celebrating our common values as a people." ---- Linda Douglass

Is it common or grammatically correct to use "as a people" here in the sentence above to describe "our common values"? But "values" are different from "a people", right?

Yes, it is both correct and often said. The "as a people" is an adverbial phrase (of manner) that modifies "common". Common in what way? As a people. For all of us as one nation.

[latenight]

Could it be possible to modify "our" as a people? But why does it use "a people" instead of "people"?...

[abaka]

But why does it use "a people" instead of "people"?...

People (no "a") = persons. plural.
We the people of the United States...

A people = a nation. Singular. This is in fact the original meaning, Latin populus.
We the people of the United States...
We the peoples of the United Nations...

(Note I've put the US constitution under both. Which one is it? I suspect the second, but I'm not completely sure.)

Could it be possible to modify "our" as a people?

Yes. Perhaps you are right. Both make sense, in terms of meaning. Parsing can be subjective! But even if "as a people" modifies "our", it is still an adverbial phrase of manner.