This was probably a libation, or ritual pouring of liquid as an offering to God/the gods (it was a common religious practice everywhere in antiquity, not just in Israel/OT).
As noted in the linked Wikipedia article, the first mention of a drink offeirng isn't in Numbers, it's in Genesis, and it specifically mentions Jacob pouring liquid on a pillar he sets up as a memorial after God changes his name to Israel:
And Jacob set up a Pillar in the place where he had spoken with him, a Pillar of Stone; and he poured out a drink offering on it, and poured oil on it.
— Genesis 35:14
It seems that Jacob didn't pour the liquid on an altar because drink offerings weren't required to be made on altars, but the pillar seems to have been related to an altar as a memorial and way to remember God. Here's an article from a Jewish perspective that has an interesting take on the distinction:
But as noted by Rabbi Abraham Isaac Kook, the first Ashkenazi chief rabbi of pre-state Israel, there is a crucial distinction between pillars and altars: the former are constructed of one large stone (matzevah, from the root meaning “to stand”), the latter of many stones (mizbeach, from the root meaning to slaughter or sacrifice). One stands erect, the other lies horizontal.
Either way it seems clear that Jacob is making a ritual offering/libation here, and that's what the drink offerings you see later in the OT are all about.
EDIT: As to what this meant to Paul, which is half of your question, he seems to have conceptualized it metaphorically as pouring out his soul in his efforts for the early Christian congregations, as the liquid poured out in the offering. This fits with his tendency to abstract away the physical ritual requirements of the Jewish law into an inner law of faith:
Therefore, there is now no condemnation for those who are in Christ Jesus, 2 because through Christ Jesus the law of the Spirit who gives life has set you free from the law of sin and death. 3 For what the law was powerless to do because it was weakened by the flesh, God did by sending his own Son in the likeness of sinful flesh to be a sin offering. And so he condemned sin in the flesh, 4 in order that the righteous requirement of the law might be fully met in us, who do not live according to the flesh but according to the Spirit.
(Romans 8:1-4, NIV)
