Genealogies may refer to one of two ideas. One is a physical bloodline. Another is a spiritual bloodline. Both are mentioned in the Bible.
Physical genealogies were in fact used to prove certain histories, relationships, and descendants. These were used to arrive at what was considered certified truth. Here are examples. They led from Adam to Jesus. They led from the first high priest Aaron to the last high priest. They led from David to Jesus. These written genealogies were necessary to positively prove position, birthright, and identity.
Spiritual genealogies are similar, showing spiritual authority from one Christian to the next. For example, Jesus instructed the apostles to preach the gospel and breathed on them the Holy Spirit. Paul lays hands on Timothy and instructs him to carry on the work. Simon the Socerer on the other hand attempts to be part of this lineage, but Peter rejects his offer of money. This “laying on of hands” is a foundational teaching (Heb 6:2).
So, we find “useful” genealogies, but to what was Paul speaking regarding the “useless” genealogies in these two verses?
Neither give heed to fables and endless genealogies, which minister questions, rather than godly edifying which is in faith: so do. 1 Tim 1:4
But avoid foolish questions, and genealogies, and contentions, and strivings about the law; for they are unprofitable and vain. 1 Titus 3:9
In context in both instances, Paul contrasts the teaching of faith with the teaching of the law. In other words, certain people were using their genealogies to prove their teaching authority, but their teaching skewed from the truth of the gospel.
PHYSICAL BLOODLINE
As mentioned, there was a time and place for proving certain physical bloodlines through historically agreed upon records. These things were written down and relied upon. When someone could not prove their history, they were rejected from certain positions (Neh 7:64).
Could genealogies be misused? Absolutely, this is to what Paul is referring.
This will be controversial to some, but this scriptural example makes the most sense within the context that Mary and Joseph indeed had children as scripture and early tradition tell us. Certainly, a claim to being the physical brother from the same mother to the Son of God would carry some weight and influence. Alternatively, this claim could also arise within the context of being a step-brother from an earlier marriage of Joseph’s. This relationship helps explain why James, as opposed to Peter or Andrew or any of the other apostles who actually believed and walked with Jesus prior to His ascension, was chosen as first bishop of Jerusalem.
Imagine the argument. Those may have walked with Jesus, but James is of His bloodline. He too has descended from David. He is an heir of the kingdom. He should be first bishop in our center of the earth.
In scripture, we find these things shown as clashes between certain parties.
For before that certain came from James, he [Peter] did eat with the Gentiles: but when they were come, he withdrew and separated himself, fearing them which were of the circumcision. Gal 2:12
And the day following Paul went in with us unto James; and all the elders were present. Acts 21:18
This incident led to an uproar. Paul was arrested. But recall that the decision regarding the Mosaic Law had already been settled, yet there it was again.
And after they had held their peace, James answered, saying, Men and brethren, hearken unto me: Acts 15:13
For many, James was the go between of heaven and earth, trying to sooth both sides of the dispute as diplomatically as possible. For Paul, there is only one gospel.
This physical bloodline led to strife, promoting speculations about God’s plan received by faith, rather than works. This continues today.
SPIRITUAL BLOODLINE
The next incident was also about the truth of the gospel; that is, we are saved by grace through faith not of our works, but of God’s. Who is the truth? Who has the truth? How do we know?
Now this I say, that every one of you saith, I am of Paul; and I of Apollos; and I of Cephas; and I of Christ. Is Christ divided? was Paul crucified for you? or were ye baptized in the name of Paul? 1 Cor 1:12-13
People were elevating certain disciples, rather than elevating the truth of the gospel. It is not who says what, but what is said by whom. The plumbline of truth is not men, but what God has said. How do we know for certain? Scripture, the “this is written”.
CONCLUSION
OP: I would guess, especially based on the context, that "certain people" were more or less fabricating geneologies about biblical figures, but on what basis?
This is true as we saw. People and their followers invent genealogies about themselves, about their histories. The basis is a claim to physical or spiritual bloodlines.
So, while some in Paul’s time were and still today are busy establishing blood or spiritual genealogies in order to prove their position, their power, their prestige, or their teachings, Paul was dismissing the spiritual and physical bloodlines as carrying any weight and admonishing instead their focus on the truth of Christ crucified and risen.