Is Divine Sonship of the Lord Jesus the Messiah His office?
The begetting is indeed not biological, it does not result from physical birth, the Father and Son are Spirit, it is mysterious & we are His analogy. However do you think the Sonship is accurately described as an 'obligation, duty or function' ascribed the Son? That is meaning of the word office.

Latin: officium, officii [N] duty, service

His priesthood, prophethood and monarchy are all mediatorial offices. Each was inaugurated in time, though fruit of an eternal decree, (Ps.110.4, Heb.7.21-2, Ps.2.6, Acts 5.31, Ps.89.27, Deut.18.15) and eternal in duration, they are assumed by the Son and appointed by the Father (Heb.5.4, 1 Chron.6.32). He becomes the Lamb, though slain before the foundation of the world, yet He has always been the Son.

“‭No man has seen God at any time; the only begotten Son, Who is (ho on) in the bosom of the Father, He has declared Him.” John 1:18

He is assigned the Name of Saviour & Messiah, Matt.1.21, Acts 2.36, but in essence, in being, He has ever been the Father's delight, Eph.1.4-6, Tit.1.2. He is anointed in time to His office, Isa.61.1, Lk.4.21, He always has been the Son, 1 Jn.4.16, Jn.15.9, 17.26.

“‭All things are delivered to Me of My Father: and no man knows Who the Son is, but the Father; and Who the Father is, but the Son, and he to whom the Son will reveal Him." Lk.10.22

The Scriptures indicate His Sonship is eternal, 1 Jn.1.2, uncreated, unassumed, not an assignment, duty or function in itself, though it entails obligation. He is the Son by nature, in His essence, not by the Father's mere appointment, but by eternally partaking in His life, Jn.6.57, 5.26, 17.5. The Son is rooted in the Sacred Name. He is not a mere attribute, title or description of the Father, but very God of very God, the truly begotten Son, not created or just assigned. He is distinguished from the Father, but fully partaking in His Being and glory, His very brightness, Rev.22.1,3, 21.22,23, 5.6.

Is the Father God also an office, rather than His Eternal Nature? This is the implication of the assertion.

It is an error, perhaps inadvertent to protect against Muslim misconception of a physical birth, but accuracy matters particularly here.