My family has had sort of the same problem. Mainly dime sized “greasy” spots on dress shirts and pants. My wife at first blamed me for spilling stuff on them, but later realized I wasn’t to blame. Clothes come out of the laundry with spots they didn’t have going in!
The only thing we can think of is that the clothes are getting mixed in with our kids’ clothes which have all kinds of stains on them, which bleed over to ours. We’ve been more careful lately separating them and haven’t noticed nearly as many of those greasy stains on the shirts.