Are the definitions of IND-CCA secure and of IND-CCA secure under standard model identical? Suppose there is a public key cryptosystem based on the discrete logarithm problem and let this cryptosystem be IND-CCA secure under standard model. Does this statement mean that the given cryptosystem is IND-CCA secure if Diffie-hellman assumption holds (that is the problem on which the given cryptosystem is based) or there may be any other assumptions (problems)?
A cryptosystem is not "based on an assumption" ; it is based on some mathematical structure (e.g. prime order elliptic curves, or prime order fields). Informally, a cryptosystem is said IND-CCA secure (which means: it satisfies the indistinguishability security notion, against adversaries which are given access to a decryption oracle) under some assumption A if given access to an adversary that breaks the IND-CCA security of the scheme, one can construct an algorithm breaking A.

But you cannot say that "A is the problem on which the cryptosystem is based": for a given cryptosystem, different security notions can be based on different assumptions. For example, take the case of ElGamal: it is IND-CPA secure under the DDH assumption, and (a slight variant of) it was also proven IND-CCA1 secure (CCA1 indicates that the adversary can make decryption queries only before receiving the challenge ciphertext - this is also known as security against lunchtime attacks), but only under a very different and less standard assumption (see this paper for example, or some of the related papers it mentions in the intro).

So when a cryptosystem is said to be IND-CCA secure, it does not tell anything about the assumption on which it is based.

The term "in the standard model" relates to a completely different things (it does not say anything either on the particular computational assumption to which a security notion for the scheme can be reduced): it is common in crypto, when dealing with a hard problem, to consider simplified "worlds" in which the players could have access to some ideal primitive, and then to instantiate this primitive "in practice" with something which seems a good candidate for this ideal primitive, based on our current understanding of it.

The most common example is that of the random oracle model: we assume that the players are in a world in which they have access to an oracle which acts as a truly random function, and then we prove that some cryptosystem is secure (e.g. IND-CCA secure) under some computational assumption in this simplified world. This does however not prove that in the real world, the cryptosystem would be secure; using SHA-256, for example, as a random oracle seems to work in practice, but SHA-256 is not a random oracle.

Saying that something is secure in the standard model is just a way to say that no such idealized world is considered: the proof is "in the real world" and does not assume any idealized primitive.