Jesus explains the fundamental principle at work in Acts 9:5 by means of what he says in Matthew 25:34-45 (KJV) -- specifically:
And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me.
-- Matthew 25:40 (KJV)
That is, for those who would claim to know God, "What they do unto others, they do unto Jesus".
When Jesus says, "I am Jesus whom you are persecuting...", Paul didn't offer an argument in his defense, "When did I persecute you?", because he immediately became aware that what he had done to Jesus' followers was no different to having done it to Jesus himself.
Paul refers to the church as being analogous to the body of Christ:
For we are members of his body, of his flesh, and of his bones.
-- Ephesians 5:30 (KJV)
He uses the analogy to explain that as the body consists of many different parts all working together for the benefit of the whole, so too, the church.
Paul's analogy is just a means of illustrating what Jesus said in John 17 (emphasis mine):
11 And now I am no more in the world, but these are in the world, and I come to thee. Holy Father, keep through thine own name those whom thou hast given me, that they may be one, as we are ...
17 Sanctify them through thy truth: thy word is truth. 18 As thou hast sent me into the world, even so have I also sent them into the world ...
20 Neither pray I for these alone, but for them also which shall believe on me through their word; 21 That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me ...
25 O righteous Father, the world hath not known thee: but I have known thee, and these have known that thou hast sent me. 26 And I have declared unto them thy name, and will declare it: that the love wherewith thou hast loved me may be in them, and I in them.
-- John 17:11,17-18,20,25-26 (KJV)
Jesus did not die, but continued/continues in the world in a NEW BODY -- the Church. Each believer is a cell in that body, so any attack on (persecution of) a member of the Church, is an attack on Jesus himself.