I experience this pretty frequently, from a couple of different angles.
Oftentimes, I’ll be so cognitively invested in a game that its systems, mechanics, and to a degree, its aesthetics will become a part of my subconscious. I find this happens most often in atmospheric puzzle games that combine abstract thought with psychological isolation — think Myst, The Room, and The Witness. For weeks while playing The Witness, I couldn’t go anywhere without noticing grid patterns in public and feeling that neural “tingle” — a sort of subconscious prompt that you’re encountering a familiar pattern, and a certain part of your brain needs to activate to try to understand it. I emphasize the aesthetic factor on this, because it only serves to amplify this effect. The deliberate quasi-steampunk styles of Myst and Riven have been hardwired into my mind, and any time I encounter complex, beautiful mechanical systems, I’m transformed back to those fond experiences of solving those puzzles, much like the effect of scent-memory when you smell a familiar aroma that’s tied to a past experience.
On another level, many games can be emotionally haunting, and I think that’s what @KestrelPi and @ChronoPunk are getting at here. Those narratives that not only tug at your heart strings, but get you invested in a way that forces you to think about how these characters’ experiences relate to your own. Horizon Zero Dawn did it most recently for me. The awe-inspiring world design — those moments where you come over a ridge and gasp at the sheer beauty and serenity of the landscape before you — drew me in and made me care about how truly human the characters felt. That level of investment stuck with me, making it so that even small emotional moments throughout the story could make me tear up, and coming across a familiar sight or concept in person filled me with a bittersweet fondness of HZD’s emotional arc.
Edit: Even more than HZD, the week I spent playing Life Is Strange was an emotional roller coaster. Every gut-punch and twist in those five episodes left me with a sense of overwhelming, profound numbness — in a good way. In a way that made me think, “Wow, these characters’ struggles are so much more real than any of the stuff I think is a problem for me.” I think, more than anything, the power of games that “haunt” you are that they create empathy in the player, and make us reconsider our perspectives from encountering emotionally taxing experiences through others’ eyes.