Kelly has a mole on the right side of her upper lip -except when she is seen as an elder. Then it switches to her left.
In 1987, when Yorkie goes to the Quagmire, the Pixies song Something Against You is playing. The song wasn't released until 1988. However, San Junipero was not primarily intended to be scrupulously free of anachronisms; as a place where the elderly and dying can go to relive fond memories of their youth, recreating the overall look, sound and feel of the era would be much more important than making sure everything was pinpoint accurate.
Yorkie is seen driving a Mazda MX-5/Miata in San Junipero. The Miata wasn't available until 1989, however there's no reason to think she hasn't decided to change the period she's experiencing to 1989 rather than 1987, not to mention the fact that the simulation isn't 100% accurate.
There are several scenes that show elder Kelly wistfully looking out from her nursing home at what looks an awful lot like the ocean by her house in San Junipero. Since her nursing home is in Carson City, Nevada- which has no coastline- this is impossible. However, it is very likely that Kelly's view was never intended to be that of an ocean, as Nevada's capital sits on the shores of another large body of water- Lake Tahoe.
When Yorkie is talking about her fiance, she says "My family don't approve," which would be the way a Briton would say that sentence (or write it in a script). If Yorkie is American, she should say "My family doesn't approve," because "family" is a singular noun. However there's no reason to think Yorkie wasn't born in England or grew up there; she may have simply emigrated with her family as a child.
Near the end, when Yorkie is driving the convertible while "Heaven Is A Place On Earth" is playing, she is driving from the left hand side of the car in all close ups, but is clearly on the right hand side of the vehicle during the aerial long shot driving down the highway.
The video arcade games are those available in the time period specified. The "Bubble Bobble" game, originally released in August 1986, in 1987 shows artwork (around the screen) with the publisher Taito's 1988 style logo. This design is fan art, and was not used in the original artwork for this game. Given the true time setting of the story, this is conceivably a genuine mistake based on inaccurate historical records and best guesses.
The town of San Junipero, while fictional, is based on the real Junípero Serra and is mispronounced. Many things are named for San Junípero in California, where the J is pronounced like an "H" as it is San Jose.