The JPS translation is:
You keep count of my wanderings;
put my tears into Your flask,
into Your record.
(Psalm 56:9, JPS, verses numbered different in MT)
Boice stated that the different ways this verse has been translated does not change the application: God is not indifferent to our cares.
This section ends with a prayer that God will judge these enemies (v. 7) and with a request that God will remember his sorrows, making a list of them (v. 8). David knows that God knows what he is going through and that he will remember it. In fact, he presents the tender concerns of God for himself and his people in an image that has been of immense comfort to generations of sorrowing believers. We know it best in the words of the King James Bible: “Put thou my tears in thy bottle.” But the idea is much the same in the New International Version: “list my tears on your scroll” or “put my tears in your wineskin” (footnote). The meaning is that God will never forget nor ever be indifferent to the cares of any one of his much-beloved people.
Boice, J. M. (2005). Psalms 42–106: An Expositional Commentary (p. 470). Grand Rapids, MI: Baker Books.
You keep count (סָפַ֪רְתָּ֫ה) and into your record (בְּסִפְרָתֶֽךָ) have the root סָפַר which has the idea of writing.
Your flask (בְנֹאדֶ֑ךָ) literally means skin, either wineskin or parchment. Wineskin makes more sense for keeping a liquid, but parchment fits the since of writing better. That's the basis of the to ways to translate this verse.
If tears in a bottle were a known custom, there wouldn't be this variation in translation. The translation would be known.