Is there a way to know who the actual manufacturer is of a store-brand over-the-counter drug?

Not that I know of. . . I went on company tour in 1998. It was through a organization I was a member of.
We went to a packaging company that did “blister packs” for pills and tooth paste tubes. They showed us the equipment and technology but not the labels for their customers.

They got the pills from another company

Well, in an attempt to answer this question I came across this

Wow, interesting. I almost always look up where my prescribed drugs are manufactured. The manufacturers name is right on the Rx bottle.

I’ve never looked for the name of the manufacturer on OTC medication. I just looked at my generic acetaminophen bottle and I didn’t see any info, although the writing is very tiny, so I might have missed it, or maybe it is on the box and not the bottle.

I googled and it doesn’t look like it is required on the packaging.

I googled more and found my Walgreens acetaminophen is made by LNK International. Looks like they make a lot of OTC. https://www.lnkintl.com/

I think by the bar code you can tell. Here is a ist of bar codes:
https://www.upcbarcodes.com/learn/barcodes-demystified/country-codes/

The link I left above shows barcodes by country. Very helpful. Country of origen is information many people want to know.