aud (Audience) ClaimAccording to RFC 7519:
The "aud" (audience) claim identifies the recipients that the JWT is intended for. Each principal intended to process the JWT MUST identify itself with a value in the audience claim. If the principal processing the claim does not identify itself with a value in the "aud" claim when this claim is present, then the JWT MUST be rejected. In the general case, the "aud" value is an array of case- sensitive strings, each containing a StringOrURI value. In the special case when the JWT has one audience, the "aud" value MAY be a single case-sensitive string containing a StringOrURI value. The interpretation of audience values is generally application specific. Use of this claim is OPTIONAL.
The Audience (aud) claim as defined by the spec is generic, and is application specific. The intended use is to identify intended recipients of the token. What a recipient means is application specific. An audience value is either a list of strings, or it can be a single string if there is only one aud claim. The creator of the token does not enforce that aud is validated correctly, the responsibility is the recipient's to determine whether the token should be used.
Whatever the value is, when a recipient is validating the JWT and it wishes to validate that the token was intended to be used for its purposes, it MUST determine what value in aud identifies itself, and the token should only validate if the recipient's declared ID is present in the aud claim. It does not matter if this is a URL or some other application specific string. For example, if my system decides to identify itself in aud with the string: api3.app.com, then it should only accept the JWT if the aud claim contains api3.app.com in its list of audience values.
Of course, recipients may choose to disregard aud, so this is only useful if a recipient would like positive validation that the token was created for it specifically.
My interpretation based on the specification is that the aud claim is useful to create purpose-built JWTs that are only valid for certain purposes. For one system, this may mean you would like a token to be valid for some features but not for others. You could issue tokens that are restricted to only a certain "audience", while still using the same keys and validation algorithm.
Since in the typical case a JWT is generated by a trusted service, and used by other trusted systems (systems which do not want to use invalid tokens), these systems simply need to coordinate the values they will be using.
Of course, aud is completely optional and can be ignored if your use case doesn't warrant it. If you don't want to restrict tokens to being used by specific audiences, or none of your systems actually will validate the aud token, then it is useless.
One contrived (yet simple) example I can think of is perhaps we want to use JWTs for access and refresh tokens without having to implement separate encryption keys and algorithms, but simply want to ensure that access tokens will not validate as refresh tokens, or vice-versa.
By using aud, we can specify a claim of refresh for refresh tokens and a claim of access for access tokens upon creating these tokens. When a request is made to get a new access token from a refresh token, we need to validate that the refresh token was a genuine refresh token. The aud validation as described above will tell us whether the token was actually a valid refresh token by looking specifically for a claim of refresh in aud.
aud ClaimThe OAuth Client ID is completely unrelated, and has no direct correlation to JWT aud claims. From the perspective of OAuth, the tokens are opaque objects.
The application which accepts these tokens is responsible for parsing and validating the meaning of these tokens. I don't see much value in specifying OAuth Client ID within a JWT aud claim.