Conflating the record in Acts 5 with the record in Acts 13 does not negate the fact that the word didōmi in Acts 5:31 denotes something to be done by him to whom it is said to be given.
23 Of this man's seed hath God according to his promise raised unto Israel a Saviour, Jesus
Acts 5:
30 The God of our fathers raised up Jesus, whom ye slew and hanged on a tree.
31 Him hath God exalted with his right hand to be a Prince and a Saviour, for to give [Greek didōmi] repentance to Israel, and forgiveness of sins.
32 And we are his witnesses of these things; and so is also the Holy Ghost, whom God hath given [Greek didōmi] to them that obey him.
From Thayer's Greek Lexicon:
6. δίδωμι is joined with nouns denoting an act or an effect; and
a. the act or effect of him who gives, in such a sense that what he is said διδόναι (either absolutely or with the dative of person) he is conceived of as effecting, or as becoming its author …
or
b. the noun denotes something to be done by him to whom it is said to be given: διδόναι τινὶ μετάνοιαν, to cause him to repent, Acts 5:31; Acts 11:18;
vs 31 – God was ready to give repentance to those who slew and hanged on a tree the Lord Jesus Christ.
vs 32 – the same Greek word [didōmi] is used in this verse whom God hath given to them that obey him ... and what were they asked to do? ... receive the repentance given by God.
.