tl;dr: No, "gifted" in the MCU is not a replacement for mutants; it's the MCU term for a "metahuman". Some of the gifted people on Agents of SHIELD are based on mutants from the comics, some are based on other non-mutant characters, but most appear to be original to the MCU.
The MCU does not have X-Men style mutants (since Fox owns those, and Marvel/ABC/Disney can't use them), but it does have genetically altered humans, in the form of the Inhumans. They are technically also mutants, just with a different type of mutation, but they'll probably never be identified using that specific term for licensing reasons.
"Gifted" is what SHIELD uses to describe any person that has abilities beyond those of a normal human. For example, they consider Bruce Banner and Steve Rogers to be gifted. Rogers explicitly identifies Scarlet Witch and Quicksilver as gifted in Age of Ultron, but in the MCU, those two are not mutants (they got their powers from the Mind Stone.)
On Agents of SHIELD, there are lots of gifted on "the Index". Some of those were likely born with their gifts, and thus are probably mutants, but others were given their powers scientifically, such as the Extremis victims.
So far, a few of the gifted people who have appeared on Agents of SHIELD are based on characters from the comics:
- Skye is Daisy Johnson, aka "Quake" from the comics. She is a mutant in the -616 universe, but not an X-gene mutant; in the MCU she is Inhuman.
- Skye's father is Calvin Zabo, who's powers come from his Hyde formula in both the comics and the MCU.
- Mike Peterson is Deathlok. There have been several Deathlok's in the comics, usually reanimated robotic zombies. Deathlok in the MCU was given powers via the Centipede Serum (a derivative of Extremis), with the cybernetics added later.
On the other hand, there doesn't appear to be any basis in the comics for any of the Inhumans we've seen so far, or the miscellaneous gifted people Cal broke out of SHIELD to help track down Daisy. There have also been short glimpses of other named and unnamed gifted people are various times, none of which seem to be based on recognizable comics characters.