How does he have those memories?

The daughter (Dr. Ana Stelline) creates memories for replicants. While she tells K that it's illegal to implant human memories in replicants, she also explains artists put a bit of themselves in their work and suggests it's why the memories she creates feel so authentic (and are in such high demand). And when she sees one of K's memories (possibly that of the wooden horse), she cries and then confirms it's real. Ana implanted her own memories or possibly fragments of them (including, or maybe exclusively, the horse memory) in replicants, and K is one of them. She recognizes K's memory as her own. That's why she cries but isn't surprised by this and says nothing else.

[Another theory that's been floating around is that K, who was pretty much a walking decoy since Ana's DNA was falsely registered as his own, was also intentionally implanted with her memories to further throw people off the scent. I'm not sure about that because 1. The line about artists injecting a bit of themselves was looped when K realizes Ana was the child all along, and 2. There's a scene with Mariette, the replicant hooker who merged with Joi to sleep with K, waking up after the fact and examining the horse, seemingly recognizing it; It hints she might have that memory as well.]

You know, these takes are really good and I have my own, which I'm not sure of anymore after reading other interpretations, but here it goes anyway.

My guess is that Ana planted the memory in K's head so that it would be a sort of trigger if he ever found any clues. Basically like someone stepping into a P.I's office and hiring him to find someone, although a lot subtler and completely on a subconscious level.

She didn't know K when she met him, although she lied to him, so maybe she was pretending.

She knew it was illegal to plant real memories in replicants and she couldn't tell him that she did it, but since she had some financial power by working as a subcontractor for Wallace, it could be a possibility that she wanted to find her dad or what happened to him after she was born.

The reason I suspect this is because 2049 is a lot more of a classic noir than the original Blade Runner and even Hampton said in a recent interview that he wasn't that much inspired by K.Dick's book and more from Raymond Chandler's novels, of which he is a fan(me too). Then, in the same interview, the other co-writer, Michael Green, said that he hadn't read any detective novels before working on this and caught up with Chandler and Hammett, who are the founding fathers of the detective genre. He even said that a few noir films were used as inspiration, although he didn't give any names, but I bet Chinatown is one of them, because I rewatched it a few days ago and I could see it had a good deal of influence on 2049.

But I also like the interpretation that she planted bits of her memory in replicants in order for them at one point break free and be their own masters. She is, after all, Morpheus in this film. She has the power to give replicants the ability to dream.

We can assume that mind reading technology is employed in the manufacturing of dreams/memories. The machine K sits in front of at Stelline's does exactly this, as does Joi technology (in order to effortlessly positively reinforce/support inner desires). Likewise, the physical tool Stelline holds is a means of capturing and manipulating the images elicited by the user's mind's eye. As K stated to Madame, the child was inserted into a work program and was unaware of its origins and parentage - this is largely true, however, Stelline knows that the story about her off-world colony parents is false. Before being placed in the orphanage and shortly later emancipated from it, she was raised by replicants who were open about their own nature, and Stelline therefore has a love for replicants, as evidenced by her caring sentiments in creating memories.

Given that Stelline's neural network is at the very least similar to that of a standard replicant, her mind essentially speaks the same language as the dream scripting machine. The memory of hiding the extremely valuable wooden horse (which bears the markings of her birth date and Rachael's date of death, and was whittled by Deckard on that date) is particularly prominent in Stelline's mind, as it pertains to her missing parents. It is part mystery, part key piece of her identity, and as such it subconsciously becomes embedded in every memory she creates (this process is eased by the similarity between the machine and her cognition). The fact that she became a memory technician is pure coincidence, although her neural patterns likely mark her as a natural at the art, which neatly explains her selection and success.

Lot's of great points and fun reads! But I don't think anyone has mentioned on how the possible real daughter who presumably illegally planted a real/bad memory; ended up in such an elite position, in a bubble, making memories.

I know she says that people put her there with everything she could possibly need except company. Ok, so a girl gets sent to San Diego to work child labor after her papers are fudged that she died, or at least is hiding there while they set her up to grow up and find a place in society as a girl to fabricate memories for replicant slaves inside a bubble. OK

And back to the point because we're supposed to answer the memory issue.

Well since it's all so vague we're left to think of the abstract possibilities.
To me the easiest assumed possibility is she indeed put some of her memory (although illegal) into at least one or possibly all replicants she was involved with. I guess because it was some strange way to cope with her own memory, or a master plan to hide her identity in replicants.

Enjoyed the movie though!!!

My theorie is that Stelline put her real memories into the replicants, so that one of them would maybe act as her surrogate to find her family. All the steps K followed lead him right to Deckard who had the answers. In the end, K united the father with his daugther, possibly setting Stelline free. So I think maybe she implanted those memories to escape her prison.

K is suppose to be the replicant generation post Nexus. He is suppose to not have the issues of the Nexus versions.

For K/Joe the "horse" memory although maybe to oblivious could be 'The Trojan Horse' memory that triggers the mental/emotional freewill.

Possibly 'code/memory' has placed in post nexus version replicants by agents of remaining nexus replicants to "hack" them from their per-programmed bondage.

Why the Replicant Blade Runner “KD6-3.7” has the natural born female replicant Stelline’s wooden horse memory – I can only speculate, but after watching the movie several times and reading some other very good theories, I have made some observations and formulated my best guess.

No replicant or human thought to ask Stelline if a memory is real or not. How would anyone know anyway without going inside her mind? The memory of an orphan is hardly copyrighted material or terribly significant. A law like that in a dystopian, corrupted society especially is most likely sort of a copyright infringement clause to keep you from copying an actual human mind into a replicant, making them too much like a certain real person, etc.

Stelline seemed pretty glib about it when K asked about using real memories. They have memories manufactured on purpose, otherwise they could just download a bunch of memories from books, movies, etc. Then you would have a replicant with memories of running through a cave with a giant rolling ball chasing him, swinging from a whip, etc. Now you’re risking a lawsuit from the estate of Stephen Spielberg, etc.

Putting one of her childhood memories in with everything is else is probably a minor thing. Very likely many replicants designed by Stelline have that memory. Especially ones who are designed to be subservient protectors – Blade Runner, etc. It evokes feelings of powerlessness, but also of protecting something special. It’s a powerful one for Stelline.

It symbolizes her father and perhaps sharing that memory with replicants is her way of sharing her sense of loss, her own loneliness. Regarding how she got where she is - I theorize she was most likely briefly adopted from the orphanage by wealthy human parents but when she developed the immune syndrome they abandoned her on earth. That all could very likely be true, therefore she has no idea she is a replicant.

She has know way of knowing K found the horse and thinks he is the natural born replicant. He just misunderstood the whole thing.

Imagine though, if you wanted to start a revolution, which in the movie is apparently coming. (Sequel?) What if the figurehead of the revolution was someone you share memories with? Some one who helped form you and make who you are? That would be a powerful leader.

The biggest clue to me is that Joe is dying on the steps in the snow outside at the end and the daughter is inside examining a snowfall scene. She clearly has a connection with Joe, possibly the ability to both read his memories and possibly to implant them.

While he may have shown her the memory of the horse, how do we really know who lived it? It may actually have been his memory if he was the decoy and she may know it was real because she has seen it before in his mind. Or it may be her memory. Who knows.