Some terminology may help you.
Real pearls are either
cultured (farmed pearls) or
natural (wild pearls, rare and very expensive, created in nature by mollusks without any human intervention.)
The pearl market has been a
cultured pearl market for the past century.
Imitation (fake) pearls are made in factories and never see an oyster! A bead (glass, plastic or sometimes shell) is dipped in a lacquer made with an extract from fish scales.
The pearls on the right appear to be knotted, real cultured pearls. The variation in the shapes and sizes makes it unlikely they are imitations.
They could be lower quality akoyas, some off round, with relatively thin nacre, which produces a milky white appearance. Akoyas are much more costly than freshwaters, and good quality akoyas would cost much more than $149.99. While vintage akoyas could cost less than recent ones, vintage akoyas are more likely to be a cream color than white. Good akoyas would have better luster than that strand also.
I think they look more like freshwater pearls, which generally have lower luster than akoyas. On the plus side, if they are freshwaters, they would be solid nacre and very durable, whereas akoyas with thin nacre would be more fragile.
Additionally, some akoyas were produced in China. Not so much today any more, but if these pearls are from 10-20 years ago, they may have originated in China. I'm just telling you that since it seems a Japanese origin seems significant given your daughter's tastes.
A simple test you can do to assure yourself they are real is to rub two of the pearls together gently. (Please don't rub them against your tooth. Teeth are harder than nacre and can scratch the pearls.) What do you feel? Real pearls feel a bit gritty when rubbed together while fake pearls feel smooth. (This test can be thrown off if real pearls are coated or if fake pearls are dirty).
If the store has a jeweler's loupe you can borrow, use it to look at the surface of the pearls. Fake pearls have a grainier surface appearance than real pearls. This photo of Renee Newman's shows a fake pearl (top) and a real one (bottom) under 10x magnification.
Renee Newman wrote a book called
Pearl Buying Guide that is now out of print but can be found at many libraries. Perhaps your library has a copy. There were 6 editions, the last one in 2017.
She has also written a new book that was just published this fall, which I highly recommend. Details:
Contents of new diamond book by Renee Newman
reneenewman.com
Finally if you would like to take the CPAA course "Pearls as One" you will learn a great deal about pearls! It is loaded with photos and videos and
the question and answer section below each chapter is very informative. It is also being continually updated. This is the website, and I will message you a code to take the course for free.
www.pearlsasone.org