Your question seems to ask whether the fruit of the Christian life and the fruit of the Spirit are one and the same. Well, since the Holy Spirit’s work is to conform your life to the model of Christ, then yes - the fruit of a Christian’s life is simply the outworking of the Spirit inside him (Philippians 2:13).
I agree with @SeanO that all Christians bear fruit - any branch that doesn’t is cast into the fire. Some may bear 30-fold, some 60, some 100, but they all bear some measure of fruit. That’s the whole point of the parable of the sower - Christians are good ground, not thorny or stony hearts.
And there are many things that the Bible mentions as fruit. As @mgaplus4 noted, one fruit of a Christian is other Christians. When Paul told the Romans in 1:13 that he wanted to have fruit among them as in other places, he was meaning converted souls. In John 4:36, Jesus described the Samaritan harvest that was beginning as “gathering fruit unto life eternal.” And while we may not all be as effective in evangelizing the lost as others, every genuine Christian has an earnest desire to see the lost saved from hell and does what he can to plant the seed, or water it, or participate in the harvest. And all who contribute to another’s salvation participate in the reward (I Corinthians 3:6-8).
All Christians are witnesses for Christ (Acts 1:8). Whether you’ll be a witness isn’t the issue - what kind of witness you are is. The moment others know you’re a Christian, you are immediately under the world’s microscope. And what they see in your life will either challenge their unbelief or justify it to themselves.
And as you yourself mentioned, the fruit of the Spirit mentioned in Galatians 5:22-23 lists nine characteristics - but it has an open ended finish - against “such” there is no law. That word “such” means the descriptions aren’t really limited to the nine alone, but all “such” virtues describe the fruit of the Spirit.
There are many other things in the Bible that are called fruit - but I’ve often said that you could just sum it all up as “anything that’s different about your life (or those you’ve influenced) because of Jesus Christ”. That’s the fruit of the Holy Spirit in your life. The difference between where you are now, and where you would be if you had never come to Christ - all of that is the fruit that the Spirit has produced in your life so far.