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I made pineapple vinegar by just putting all of the fibrous and hard to eat parts of the fruit in water with some sugar. It did ferment and make vinegar, in that it's sour and pretty acidic (pH is between 3 and 4, in my testing), but it's a bit harsh and not very subtle in flavor.
How can I age or rest or store this vinegar to get something pleasant for eating or cooking?
As Unlisted said, real balsamic is indeed matured in elaborate ways, using storage in wood. But most of the cheaper stuff you can buy is not made that way.
When you buy a very tasty fruity vinegar, it is usually just a mixture of actual vinegar and fruit syrup. Sometimes also fruit juice, if you want to keep more of the sour taste. This is very easy to achieve - just get some fruit syrup or juice, maybe adding some other sweetener like beet syrup, and add some to the vinegar. Work on some minimal test batches first, and start with very low concentrations, until you know your preferred mixture - you don't want to make it too sweet all at once.
If you want to try aging, you don't have to buy an actual barrel. Even in industry, it is common practice to use wood chips inside of nonreactive vessels - and you can dose the chips for any batch size. Just make sure you get untreated wood.
An alternative way to infuse taste, combining both of the above, is to take ripe fruit, cover it in vinegar, and let it stay for a couple of weeks. The vinegar will take on the taste.
Even 'fish & chip' malt vinegar is matured in wood for a week [some cheap ones only a day].
https://www.sarsons.co.uk/how-vinegar-is-made
Balsamic is nurtured like a fine sherry for years through successive wooden barrels, each barrel being fed successively from younger batches, never being entirely emptied.
https://www.seriouseats.com/everything-you-need-to-know-guide-to-balsamic-vinegar
This would feel a tad excessive for a simple experiment & limited quantity.
Wine will mature further in the bottle, but spirits won't.
By extrapolation…
I'd take the chance that without access to a maturing barrel [again excessive for a one off test] then you could just try keeping it in the bottle for a week, or a month.
It shouldn't go 'off', it's vinegar.
