Question: What does the phrase "I in the Father and the Father in me" mean in John 10:38 as well as John 14:10?
It is unnecessary, and even imprudent, for us to invent an explanation--when Jesus explains it--in detail, even using metaphors. Additionally, John goes on to expound on this, in 1 John 2.
Answer: Jesus' multiple explanations, as well as John's own explanation, affirm that this statement speaks to "Unity", the Unity between himself and his Father, the Unity that the Church is supposed to have, and the Unity the believers are supposed to have in Jesus, and the Father.
NOTE 1: The currently accepted answer, and even the Question, both omit the full contexts, (the entirety of John 14, 15, 17, and 1 John 2)--thereby omitting Jesus' own explanation, and John's explanation in 1 John 2.
Question Restatement: What does the heavily used figure of speech mean: "In that day you will know that I am in My Father, and you in Me, and I in you," in John 14:20, and the following Metaphor in John 15:5, "4 Abide in Me, and I in you. As the branch cannot bear fruit of itself unless it abides in the vine, so neither can you unless you abide in Me."
Wrongful Presuppositions: As seen in other answers, doctrinal presuppositions are wrongfully injected, obstructing valid exegesis of the text--that is, doctrines of: Trinity, Oneness, or Unitarian doctrines, (whether Jesus is the Most High God, or actually the Father, etc).
"One", in Greek, does not always denote, "the number 1", but also is a "generic pronoun," and the word for "Unity", ("εἷς" (masculine), "μία", (feminine) or "ἓν", (neuter)).
It is easily inferred from the Context, whether "One," connotes: the "Number 1"; the "general pronoun, "One"; or a "Unity".
The connotation of the "Number", is very apparent from the context:
Matthew 5:29 -
GRK: ἵνα ἀπόληται ἓν τῶν μελῶν
NAS: for you to lose one of the parts
Matthew 5:41, Matthew 25:24, etc.
Jesus, Himself, Explained: Abiding in Another, Means Unity
"One, in Unity" (Neuter), is used exclusively in all contexts illustrating "Unity", and there is nothing in these contexts to indicate a "Numerical 1", but rather groups.
Because it is obvious that, "they may be one" does not connote, "a single, one, person", then it is a certainly true that "just as we are one", does not connote a number--but rather the concept of Unity :
John 17:20-23, NASB - “I do not ask on behalf of these alone, but for those also who believe in Me through their word; that they may all be one[ἓν]; even as You, Father, are in Me and I in You, that they also may be in Us, so that the world may believe that You sent Me. The glory which You have given Me I have given to them, that they may be one[ἓν], just as We are one[ἓν]; 23 I in them and You in Me, that they may be perfected in unity[ἓν], so that the world may know that You sent Me, and loved them, even as You have loved Me.
Jesus and John State Unity/Abiding in Another is Confirmed by the Work
John 14:10, NASB - 10 Do you not believe that I am in the Father, and the Father is in Me? The words that I say to you I do not speak on My own initiative, but the Father abiding in Me does His works. 11 Believe Me that I am in the Father and the Father is in Me; otherwise believe because of the works themselves. 12 Truly, truly, I say to you, he who believes in Me, the works that I do, he will do also; and greater works than these he will do; because I go to the Father.
John 15:5 - I am the vine, you are the branches; he who abides in Me and I in him, he bears much fruit, for apart from Me you can do nothing.
John 15:9-10, NASB- Just as the Father has loved Me, I have also loved you; abide in My love. If you keep My commandments, you will abide in My love; just as I have kept My Father’s commandments and abide in His love.
1 John 2:4-6, NASB- 4 The one who says, “I have come to know Him,” and does not keep His commandments, is a liar, and the truth is not in him; 5 but whoever keeps His word, in him the love of God has truly been perfected. By this we know that we are in Him: 6 the one who says he abides in Him ought himself to walk in the same manner as He walked.
1 John 2:24 - As for you, let that abide in you which you heard from the beginning. If what you heard from the beginning abides in you, you also will abide in the Son and in the Father.