TL;DR: Paradoxically, that it is impossible to lose one's salvation is a belief supported by both sides of the issue.
One can clearly fail to receive God's promise:
For whoever gives you a cup of water to drink in My name, because you belong to Christ, assuredly, I say to you, he will by no means lose his reward.
— Mark 9:41
Look to yourselves, that we do not lose those things we worked for, but that we may receive a full reward.
— 2 John 1:8
“Behold, I am coming quickly! Hold fast what you have, that no one may take your crown.
— Revelation 3:11
But once one has received salvation, it will not be taken back:
And I give them eternal life, and they shall never perish; neither shall anyone snatch them out of My hand.
— John 10:28
There shall be no night there: They need no lamp nor light of the sun, for the Lord God gives them light. And they shall reign forever and ever.
— Revelation 22:5
The difference in doctrine results from the understanding of when one receives salvation.
The once-saved-always-saved people believe that salvation occurs when one is "born again" at baptism and the reception of God's holy spirit.
The above, and other similar scriptures, are simply examples of people that were never truly saved in the first place,
and who only thought they were saved, or even pretended to be saved.
The apparent "loss of salvation" is simply an obvious sign of something that was inevitable for such people; they were not among those that had received salvation.
Those that are truly saved will not exhibit such behaviour.
Other denominations believe that salvation is not received until Christ's return, at which time the saved are converted into immortal spirit beings:
in a moment, in the twinkling of an eye, at the last trumpet.
For the trumpet will sound, and the dead will be raised incorruptible, and we shall be changed.
— 1 Corinthians 15:52
It is then that the combined human and divine spirit, which has been developing within each true Christian since Baptism, is "born again":
Jesus answered and said to him, “Most assuredly, I say to you, unless one is born again, he cannot see the kingdom of God.”
… That which is born of the flesh is flesh, and that which is born of the Spirit is spirit.
… The wind blows where it wishes, and you hear the sound of it, but cannot tell where it comes from and where it goes. So is everyone who is born of the Spirit.
— John 3:3,6,8
Being "born again" does give one immortality and salvation, but that final process happens when Jesus returns, not during baptism.
Being a true Christian isn't enough though.
Receiving God's holy spirit at baptism is necessary for salvation, but it is by no means sufficient.
Occasional slips, even involving serious sins, are possible, but one must continue strengthening one's character right up until the end (either one's death, or Christ's return):
For you have need of endurance, so that after you have done the will of God, you may receive the promise
— Hebrews 10:36
And you will be hated by all for My name’s sake. But he who endures to the end will be saved.
— Matthew 10:22, Mark 13:13
but Christ as a Son over His own house, whose house we are if we hold fast the confidence and the rejoicing of the hope firm to the end.
— Hebrews 3:6
For we have become partakers of Christ if we hold the beginning of our confidence steadfast to the end
— Hebrews 3:14
And he who overcomes, and keeps My works until the end, to him I will give power over the nations
— Revelation 2:26
Paul himself had no doubt that he was a baptised Christian, that he carried God's holy spirit within him, or that it was his purpose in life to perfect his character.
But even he had concerns about the possible result of failing his duty as a Christian:
But I discipline my body and bring it into subjection, lest, when I have preached to others, I myself should become disqualified.
— 1 Corinthians 9:27