In Romans 5:15-16, there are actually three different Greek words that are rendered with the word “gift” in English: charisma, dorea, and dorema.
Romans 15-16 ESV (brackets added)
15 But the free gift [charisma] is not like the
trespass. For if many died through one man’s trespass, much more have
the grace of God and the free gift [dorea] by the grace charis of that
one man Jesus Christ abounded for many. 16 And the
free gift [dorema] is not like the result of that one man’s sin. For
the judgment following one trespass brought condemnation, but the free
gift [charisma] following many trespasses brought justification.
Charisma (Strong’s 5486) means a gift of grace, with emphasis on the grace or graciousness of God, who is understood to be the giver of the gift. Thayer’s provides a definition that reflects how charisma is used in the context of Romans 5.
Thayer’s Greek Lexicon
the economy of divine grace, by which the pardon of sin and eternal
salvation is appointed to sinners in consideration of the merits of
Christ laid hold of by faith, Romans 5:15
Dorea (Strong’s 1431) means a (free) gift, the emphasis being on the fact that it is free. According to Vine’s, it is always used in reference to a spiritual or supernatural gift.
Vine's Expository Dictionary
denotes "a free gift," stressing its gratuitous character; it is
always used in the NT of a spiritual or supernatural gift
Dorema (Strong’s 1434) means simply gift or bestowment. Note that there are only two occurrences of dorema in the NT (Rom 5:16 and Jam 1:17).
Thayers’ Greek Lexicon
a gift, bounty benefaction; Romans 5:16; James 1:17.
In Rom 5:15, charisma is the word used to preface the contrasts that follow between God’s gift and man’s trespass. Per Thayer’s Lexicon, charisma is the “economy of divine grace” by which men are pardoned and saved. In answer to the OP’s question, charisma is the gift of God (cf 6:23) for the purpose of man’s redemption. It functions as the overarching concept, encapsulating the essential elements of that work.
Based on the above definitions and the context of Romans 5:15-17, those elements are understood to be: the grace [charis] of God, the gift [dorea] that came by the grace [charis] of Jesus Christ, and one other gift [dorema]. Dorema is viewed as distinct given the distinctness of the word itself. However, the participial construction is ambiguous, as evident in the variation seen among the English translations (Rom 5:16 Parallel Translations). Here are two examples:
Romans 5:16a
And the gift is not like that which came through the one who sinned. –
NKJV
And not as through one who sinned [is] the free gift – LSV
Of the two, my preference is for the LSV (Literal Standard Version). Compare it to the pared down interlinear below (interlinear):
καὶ οὐχ ὡς δι᾽ ἑνὸς ἁμαρτήσαντος τὸ δώρημα
And [is] not as through one having sinned the gift
In my opinion, a problem with the NKJV and other English translations regards the addition of the article to the participial construction. The phrase “one who sinned” says something very different from “the one who sinned.” Whereas “the one who sinned” implies a particular sinner, “not as through one who sinned” simply denotes a person who has not sinned. Though it is worded differently, the ESV’s “that one man’s sins” is similar to the NKJV in connotation.
According to the LSV, the gift [dorema] is not through one who has sinned. While this may include Adam, the one who sinned, it is not exclusive to him. Comparing the use of dorema in Rom 5:16 to that of Jam 1:17, the idea of the gift [dorema] being through one who has not sinned aligns well with the “perfect gift” [dorema] in Jam 1:17.
Presumably, the gift is through one who is without sin, Jesus Christ (cf 2 Cor 5:21). With this interpretation in mind, note that the gift [dorema] lies between two parallel statements in what almost appears to be a chiasmic structure. The center of a chiasm is always important, and dorema sits in that position of prominence here.
Romans 5:15-17 ESV (tabs added)
But the free gift [charisma] is not like the trespass.
For if many died through one man’s trespass, much more have the grace of
God and the free gift [dorea] by the grace of that one man Jesus Christ
abounded for many.
And the free gift [dorema] is not like the result of that one man’s sin.
For the judgment following one trespass brought condemnation,
but the free gift [charisma] following many trespasses brought justification.
For if, because of one man’s trespass, death reigned through that one man,
much more will those who receive the abundance of grace and the free gift
[dorea] of righteousness reign in life through the one man Jesus Christ (LSV).
Looking at the composition alone, we can see how dorema serves as a bridge between the two statements. In the phrase “not as through one who sinned,” the word “through” is therefore key. Note how “through”(Strong’s 1223) appears again at the end of v17 in “through the one - Jesus Christ” (LSV).
Returning to the OP’s question, charisma is the overarching concept within which dorema plays a key role. While the nature of the gift is not specified, everything in the context suggests that the gift [dorema] is Jesus Christ, who alone without sin can serve as the mediator and instrument of God’s gift of salvation.