In NIST modulus and key size recommendations is `group` equivalent to modulus and `key` equivalent to exponent?
Yes, when using cryptosystems based on the difficulty of the Discrete Logarithm in $\Bbb Z_p^*$ or a subgroup thereof, a key size recommendation of a 3072-bit group means that the modulus $p$ is 3072-bit, and a 256-bit key implies that a private exponent (a key) is 256-bit.

In some (as far as I know: all) NIST contexts, a 256-bit key additionally implies that the order of the public generator $g$ is a 256-bit prime $q$, and (thus) that $p-1$ has that 256-bit prime factor $q$, and public keys belong to a subgroup of $\Bbb Z_p^*$ having order $q$. The generation of $p$ and $q$ is described in FIPS&nbsp;186-4 appendix A (Dave Thompson's comment explains that's for various sizes including 3072/256).

In all contexts, a private exponent should be at least twice as wide as the security level in bits, which guards against Baby-step giant-step and Pollard's rho. It is good and customary that $p-1$ has a large prime factor $q$, and that the order of the generator $g$ is a multiple of that $q$. That $q$ should be at least twice as wide as the security level in bits, which guards against Pohlig-Hellman.

The width of $p$ is dictated by resistance to a Discrete-Log extension of GNFS, which is why recommendations for this parameters match those for RSA public moduli.

In Diffie-Hellman, it can (but need not) be chosen $p$ such that $q=(p-1)/2$ is prime, and thus $g$ of order $(p-1)/2$ or $p-1$, even when the key is narrower.