It might be helpful to reflect on how the love of God is not the same kind of love that humans naturally have for their children, their spouses, their relatives, their closest friends. Humans can have such love, even if they don't believe any God exists.
So, what is this love that is unique to those who love the only true God?
This text shows that is given to some by the power of the Holy Spirit:
"And hope maketh not ashamed, because the love of God is shed abroad
in our hearts by the Holy Ghost which is given to us." Romans 5:5 A.V.
It does not say "love for God" but "love of God", suggesting that it is God's love that is given to those who are indwelt by the Holy Spirit. That would harmonize with the text that says God loved us first, before we ever loved him - 1 John 4:8-11
The process (if that's not the wrong word to use) is first, God's love for us, shown supremely in his initiative in sending his Son to die for our sins, and us being deeply moved by that, in confession, faith, and in loving Christ. As Jesus explained to his followers:
"If a man love me, he will keep my words: and my Father will love him,
and we will come unto him, and make our abode with him [via the
indwelling Holy Spirit vs. 26].
He that abideth in me, and I in him, the same bringeth forth much
fruit: for without me ye can do nothing...
Herein is my Father glorified, that ye bear much fruit." John 14:23 &
15:5, 8 K.J.
This shows how Father, Son, and Holy Spirit enable us to receive the love of God, and for that gift of love to grow in us as we abide in the Son, producing the fruit of love. All of this is done via the working of the Holy Spirit so it can be seen how this love is the love of God shed abroad in the hearts of some (not all) and such divine love produces fruit to the glory of God. The Holy Spirit being the one who produces such fruit, it can truly be said to be a fruit of the Spirit, just as much as he gifts it to us in the first place.