In NIST modulus and key size recommendations is `group` equivalent to modulus and `key` equivalent to exponent? Looking at the following NIST recommendations for a discrete logarithm, for 2016-2030 and beyond they list 3072-bit number for the group and 256-bit for the key.

If using Diffie-Hellman, does group equal the recommended size of the prime modulus and key equal the recommended size of the private key 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.