If Each is a Person in His own right, why did the Lord Jesus (the Logos made flesh) speak as though the Father and He are the same Person?

Jesus saith unto him, Have I been so long time with you, and yet hast thou not known me, Philip? he that hath seen me hath seen the Father; and how sayest thou then, Shew us the Father? (John 14:9)

The Lord Jesus spoke as though the Father and He are the same Person because the Father always dwells in the Lord Jesus and it is by the Father that the Logos acts and speaks. It is not that Jesus is the Father or is another name for the Father.

Believest thou not that I am in the Father, and the Father in me? the words that I speak unto you I speak not of myself: but the Father that dwelleth in me, he doeth the works. (John 14:10)

Are the Father and the Logos two distinct "Men"?

Yes, They are—although only the one Man, Jesus, could be seen at the time the Gospels were written.

It is also written in your law, that the testimony of two men is true. I am one that bear witness of myself, and the Father that sent me beareth witness of me. (John 8:17,18)

Can two distinct persons ever be one? Yes. Adam and Eve were distinct persons, yet they are referred to as one person.

Male and female created he them; and blessed them, and called their name Adam, in the day when they were created. (Genesis 5:2)

God "called their name Adam." The man and his wife were "one flesh."

Therefore shall a man leave his father and his mother, and shall cleave unto his wife: and they shall be one flesh. (Genesis 2:24)

The saints are destined to become one as the Son and the Father are One.

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 thou hast sent me. (John 17:21)

The saints are invited to live by the Lord Jesus just as He lives by the Father.

As the living Father hath sent me, and I live by the Father: so he that eateth me, even he shall live by me. (John 6:57)

The Father, the Logos, the spiritual environment, and the angels. It came into the Father's mind to create a spiritual world and a physical world, and spiritual creatures and physical creatures. The Father created all things through the Logos. Nothing was created that was not created through the Logos.

In the beginning was the Word [Logos], and the Word was with God, and the Word was God. The same was in the beginning with God. All things were made by him; and without him was not any thing made that was made. (John 1:1-3)

This is the first of the two beginnings. The Lord Jesus is the beginning of the first creation of God. The phrase "in the beginning" does not mean in the beginning of God, for God has no beginning. Rather, the phrase refers to the period when the first spiritual and physical creations were brought into being.

Who [the Logos, the Lord Jesus] is the image of the invisible God, the firstborn of every creature: For by him were all things created, that are in heaven, and that are in earth, visible and invisible, whether they be thrones, or dominions, or principalities, or powers: all things were created by him, and for him: And he is before all things, and by him all things consist. (Colossians 1:15-17)

To be continued.