The Greek noun, ἐξουσία (exousia), has the following meaning according to BDAG (where seven meanings are given):
3. the right or control or command, authority, absolute power, warrant, eg, Acts 26:12, 24, 27, Mark 11:28, 29, 33, Luke 20:2, 8, etc.
This meaning (by BDAG) includes Luke 5:24 - the text does NOT say of that authority is innate or delegated. However, the fact that such authority to forgive sins is nowhere delegated to humans suggests that Jesus' authority to forgive is innate.
APPENDIX - Human's forgiving sins???
Matt 16:19 & 18:18 is sometimes quoted by some who claim they have delegated authority to forgive sins on earth. Let us examine this important passage. The Greek is Matt 16:19 and Matt 18:18 is unusual. Let me quote my very literal translation.
Matt 18:18, “Truly I say to you, whatever you bind on earth will have
been bound [simple future + perfect participle passive] in heaven; and
whatever you loose on earth will have been loosed [simple future +
perfect participle passive] in heaven.”
Note these comments (in an appendix) of J B Phillips in his translation of the New Testament in Modern English:
Matthew 16:19 and 18:18, “forbidding” and “permitting”. There is a
very curious Greek construction here, viz, a simple future followed by
the perfect participle passive. If Jesus had meant to say quite
simply, “Whatever you forbid on earth will be forbidden in Heaven”,
can anyone explain why the simple future passive is not used? It seems
to me that if the words of Jesus are accurately reported here, and I
have no reason to doubt it, then the force of these sayings is that
Jesus’ true disciples will be so led by the Spirit that they will be
following the heavenly pattern. In other words what they “forbid” or
“permit” on earth will be consonant with the Divine rules.
The authority delegated here extends only as far as it accords with the will of heaven. Further, in this passage, authority is given to resolve disputes and “wrongs” between members of the Christian community. (v15-17) Again, this can only be done using the principles of Scripture under the enlightenment of the Holy Spirit.
The passage in John 20:23 conveys the same message - what we do on earth should reflect the divine will in heaven as per Jesus' model prayer in Matt 6:10, "You will be done on earth as it is in heaven." The same is true of forgiving others.