I have had a plant-based cheese company for a while. Yes, adding casein (I think you mean casein, not lactose) to soy milk woulnd't be vegan. There are companies that try to create vegan casein with precision fermentation, but to my knowledge, none of them are doing it at scale (i.e. I wouldn't know where to buy such a vegan cheese).
Most cheese is not made with acid but with enzyme called rennet, which is typically found in stomachs of cow's for example. Now, there is synthetic rennet which is used in the cheese industry widely, which is vegan. However, rennet doens't work with plant-based milks. We've done a lot of experiments to find a replacement enzyme for rennet that would coagulate soy milk with a similar result as animal-based cheese, but didn't get any great results.
The cheeses made with acid are usually softer (e.g. ricotta). For a while we manufactured and sold a plant-based cheese made with soy milk that was very similar to ricotta. Not bad, but a far cry from the animal-based version.
I still think it's possible to find a way to make plant-based cheese without needing to synthetically create vegan casein. As other commenters have mentioned, many of the vegan cheeses currently available are based on either coconut oil, cashew, or almonds (or a mix of these). For some use cases, this works suprisingly well (e.g. if you live in Germany, get the vegan Feta from Edeka's own brand. It's the fucking best). But then again for some use cases, they don't (e.g. the melting behavior is usually not very good). I think they've gotten a lot better in the past 5 years and I don't miss animal-based cheese anymore as a vegan. I also think there's a local maximum here with these coconut oil, almond and cashew based cheeses though.
Edit: Confused lactose with casein, updated to reflect that