Well we can show the Fathers declarations in which He alone states His Spirit. Jesus does speak of that Spirit as another. So that is not assumed nor mistaken that is the Fathers Spirit per His statements. Regardless if you see the Spirit as a distinct person.
Zech 4:6
So he said to me, “This is the word of the Lord to Zerubbabel: ‘Not by might nor by power, but
by my Spirit,’ says the Lord Almighty
The Fathers Promise -In the last days I will pour out
My Spirit.....Acts 2:17
And while the Spirit is in believers its stated our fellowship is with the Father and Christ Jesus.
Matt 12:18
“Here is my servant whom I have chosen,
the one I love, in whom I delight;
I will put
my Spirit on him,
and he will proclaim justice to the nations.
Nothing in regard to a person such as God the Spirit
Now this is eternal life: that they know you, the only true God, and Jesus Christ, whom you have sent.
Doesn't appear to be any ownership associated with the Spirit either. Otherwise the Spirit would take from what is His.
John 16:14-15
He will glorify me because it is from me that he will receive what he will make known to you.
15All that belongs to the Father is mine. That is why I said the Spirit will receive
from me what he will make known to you.
Jesus didn't pray to the Holy Spirit nor teach others to ask Him anything and though Mary conceived by the Holy Spirit Jesus didn't call the Spirit His Father.
Your spirit is not another person from you but it is spoken of how it knows your thoughts.
1 Corinthians 2:11
For who knows a person’s thoughts except their own spirit within them? In the same way no one knows the thoughts of God
except the Spirit of God. (Not God the Spirit)
Each of the persons of the Trinity has different roles in the salvation of humans and redemption of creation. The Holy Spirit does numerous things in creation and believers' lives, but never points to himself. He always points to the Father or the Son. It is through the Holy Spirit that the Son and Father come into believers to dwell.
Here is a list of at least some (perhaps most) of the verses in the NT which clearly show the personal agency of the Holy Spirit:
Acts: Matt 4:1; Acts 8:39, 16:7
Listens: John 16:13
Speaks: John 16:13-15; Acts 1:16, 10:19, 11:12, 11:28, 13:2, 15:28; 1 Tim 4:1; Heb 3:7
Can be lied to: Acts 5:3, which is the same as lying to God (5:9)
Bears witness: Rom 8:16; Heb 10:15; 1 John 5:6
Helps, intercedes, and searches: John 14:16, 15:26, 16:7; Rom 8:26-27; 1 Cor 2:10
Teaches: Luke 12:12; John 16:13; 1 Cor 2:13
Gives gifts: Acts 20:28; 1 Cor 12:11; Heb 2:4
Leads: John 16:13; Gal 5:18, Heb 9:8
Can be grieved: Eph 4:30
Can be outraged: Heb 10:29
Can be blasphemed: Matt 12:31-32
Convicts: John 16:8-11
These are all actions of personal agency. To expand on a few of them:
Joh 14:16 And I will ask the Father, and he will give you
another Helper, to be with you forever,
...
Joh 14:26 But the Helper, the Holy Spirit, whom the Father will send in my name,
he will teach you all things and
bring to your remembrance all that I have said to you. (ESV)
That Jesus refers to the Holy Spirit as "another Helper [Advocate; Comforter; Counselor]" is enough to show that the Holy Spirit is a person and not the Father. Teaching and bringing to remembrance are actions of personal agency. An advocate can only be a person who advocates on behalf of another:
1Jn 2:1 My little children, I am writing these things to you so that you may not sin. But if anyone does sin, we have an
advocate with the Father, Jesus Christ the righteous. (ESV)
We can see that Jesus is one advocate and the Holy Spirit is "another," as Jesus said. If we use one of the other definitions of
parakletos, we see that only persons are helpers, comforters, or counselors.
Joh 15:26 “But when the Helper comes, whom I will send to you from the Father, the Spirit of truth, who proceeds from the Father,
he will bear witness about me.
Joh 15:27 And
you also will bear witness, because you have been with me from the beginning. (ESV)
Here, the "person" who is the Helper will "bear witness" about Jesus, in the same way that the disciples will "bear witness." These are also actions of personal agency.
Joh 16:7 Nevertheless, I tell you the truth: it is to your advantage that I go away, for if I do not go away, the Helper will not come to you. But if I go, I will send him to you.
Joh 16:8 And when he comes,
he will convict the world concerning sin and righteousness and judgment: (ESV)
Notice that it is the Holy Spirit that "will convict the world concerning sin and righteousness and judgment." The Greek word for "convict" is
elegcho, and means to "reprove," "admonish," "convince," and "refute." Again, these are actions of personal agency.
Joh 16:13 When the Spirit of truth comes,
he will guide you into all the truth, for he will not speak on his own authority, but
whatever he hears he will speak, and he will declare to you the things that are to come.
Joh 16:14
He will glorify me, for he will take what is mine and declare it to you. (ESV)
Here, the Holy Spirit will guide believers by speaking whatever he hears, declaring "the things that are to come." Guiding, hearing, speaking, and declaring are also all actions of personal agency.
The Holy Spirit also goes by other names, most notably the Spirit of Christ (Rom 8:9), the Spirit of his Son (Gal 4:6), the Spirit of Jesus Christ (Phil 1:19), and the Spirit of Christ in 1 Pet. 1:11. The last one is important because it is referring to the Holy Spirit in the OT.
It is also worth noting that blasphemy against the Holy Spirit is worse than blasphemy against the Son.
Throughout the NT, the Holy Spirit is always made distinct from not only the Son, but from the Father, which is utterly pointless if the Holy Spirit is really just referring to the Father.