In the Binitarian view is the Holy Spirit a person?

In Biblical Binitarianism, promulgated by Mario Shepard, References to the Holy Spirit are sometimes references to the Spirit of the Father, sometimes the Spirit of the Son, and there is a single Spirit that they share.

According to his 22 Theses of Biblical Binitarianism Thesis number 15:

... We deny that the Bible asserts or assumes that the one spirit of the Father and the Son is another divine person, distinguishable from both the Father and the Son. We deny that any passages necessitate the nuanced concept that the spirit of the Father and the Son is someone other than the Father and the Son. ...

I interpret the above to indicate that scriptural indications that the Holy Spirit is a person are taken as scriptural references to the personhood of the Father or Son

In forms of Binitarianism where the Holy Spirit is the power of God, it's clear the Holy Spirit is not take to be a person (as found in the other answer here: https://christianity.stackexchange.com/a/93238/60459

In Binitarianism is the Holy Spirit a person?

By definition of binitarian, the Father and the Son are the two supreme beings. So the question is really asking whether "the Holy Spirit" refers to a third, but lesser, being.

If one removes the blatant bias that appears in translations of the Bible, in particular:

• The capitalization of "holy spirit" ^[1].
• The use of masculine personal pronouns ("he", "him", "his") rather than neuter ("it", "its") ^[2].
• The Johannine Comma ^[3].

the resulting scripture, unbiased by the ideas of the translators, provides effectively nothing from which one could deduce the existence of a third person, much less one that is equal to the Father and the Son.

Instead, the term "holy spirit" could refer directly to the Father and Son, as they are each holy and are each spirits, but more commonly, it simply refers to the medium by which the Father and Son interact with humans and other parts of the physical world.

At baptism, the human spirit (which distinguishes us from other animals by providing self awareness and free will (Job 32:8)) combines with some of God's holy spirit to create an embryonic spirit being that someday can be reborn as a full spirit being (John 3:7–8), a literal child of God.

For binitarians (and unitarians), the concept of a trinitarian God, whether as a simple triumvirate of three beings, or as the mystery taught by the Catholic Church, is simply one example the Christianization of a pagan idea.

For trinitarians, the concept was revealed through the traditions of the Church, the many scriptural references that could be used to support this idea do support it, and the translators' use of capital letters and personal pronouns is appropriate.

[1] Capitalization

All ancient Greek manuscripts were written with a monocase alphabet. Capitalization was supplied by the translators.

[2] Pronouns

Greek pronouns are masculine or feminine (as in French, Spanish, etc.), or neuter. This gender is associated with the word itself, not with what the word represents.
For instance, the French and Spanish words for chair (la chaise and la silla) and desk (le bureau and el escritorio) are feminine and masculine, but it would be wrong to translate pronouns referring to these objects in English as "she" and "he".

[3] Johannine Comma:

In 1 John 5.7–8 there appears a striking reference to the Catholic doctrine of the Trinity. "For there are three who bear witness [in heaven: the Father, the Word, and the Holy Spirit; and these three are one. And there are three that bear witness on earth]: the Spirit, and the water, and the blood; and these three are one" (CCD translation). The bracketed phrases appear in the Clementine-Vulgate version of the Bible, the official version of the Sacred Scriptures for the Latin rite of the Church. Among scholars these phrases are commonly called the "Johannine Comma." On the basis of manuscript evidence scholars seriously question their authenticity.

The Comma is absent in all the ancient Greek manuscripts of the New Testament with the exception of four rather recent manuscripts that date from the 13th to 16th centuries. The Comma is lacking in such ancient Oriental versions as the Peshitta, Philoxenian, Coptic, Ethiopic, and Armenian. While the majority of the Latin manuscripts of 1 John do contain the Comma, the earlier and better manuscripts, both of the Old Latin and Vulgate versions, lack it. The earliest manuscript in which it appears dates from the 9th century.

— Johannine Comma | Encyclopedia.com