you really should be able to save
Currently the demo serves as a way to experience the basic flow of the game for a few game days, starting on Monday and ending on a Saturday.
Active content wise there is one activity minigame available (cleaning) and some functionality implemented for the night session touch part to give a look at how that will work, mostly the eye inspect interactions as well as some of the hand interactions once the hand is unlocked. There's a few sequence end paths here that result in different dialogue either from the player character or from her in the morning dependent on actions, and even a bad end if some of the actions keep getting repeated, ie if you wake her up while she is slightly undressed over multiple nights. (This is the section of the game I am currently focused on for new content, primarily making new popup assets so that the other touch modes can be unlocked and new interactivity added).
Story wise the various activities have their own dialogue sequences with some varying default states once all of them have been seen. There's also a few key dialogue events spread around that will eventually function for story progression but at the moment are just for exposition purposes.
Some dialogue events only unlock when you have seen others, for example there is a special event to invite her to watch TV with you, but it only works if you have seen another unique memory event when watching TV by yourself. On a similar note certain actions will slightly change other dialogues, for example if you clean she will comment on it when she comes home from work and you will get some affection points, and if you wake up early to say goodbye enough times she will begin commenting on that as well (dependent on how many times you have done so, or even if you have overslept and interrupted your waking up early streak to various degrees).
If you cook one of the food dishes enough times it will level up, changing the dialogue that she gives when you cook it for her, and if you give her one of the three star dishes you will unlock a special dialogue related to that dish as well as a one time big boost to affection points. (You can also unlock one extra food option if you happen to catch the cooking show on the TV, later I want to add more ways to unlock food).
In conclusion I would say the demo at the moment is just enough content to get a sense for the mood of the game, experience how the dialogue can branch, and to showcase how the different interactive parts will play.