Is the Keccak family of sponge constructions the same as SHA-3? I can't seem to come across a clear and to the point answer. Is SHA-3 in the same situation as ElGamal and DSA, where it is modified slightly, or is the original Keccak submission the exact same as NIST standardized SHA-3?
SHA-3 is defined by FIPS 202.
If you go to the team Keccak website you will read the following.

Keccak is a versatile cryptographic function. Best known as a hash function, it nevertheless can also be used for authentication, (authenticated) encryption and pseudo-random number generation. Its structure is the extremely simple sponge construction and internally it uses the innovative Keccak-f cryptographic permutation.
After its selection as the winner of the SHA-3 competition, Keccak has been standardized in 3GPP TS 35.231 for mobile telephony (TUAK), and in NIST standards FIPS 202 and SP 800-185. Consequently, it has received extensive public scrutiny and third-party cryptanalysis.
[source]

Thus Keccak[1600] is the exact same as NIST standardized in SHA-3.