You have to use your imagination a little bit with this demo to see how it can support RE mechanically.
First, acknowledge what earlier RE games were. Rooms, connected by logical "breaks" in the form of doorway transitions. Mechanically, and technologically, this demo breaks those seams apart -- when you open a door here, they've lovingly brought along the feeling of a transition, but it all occurs in real-time.
We of course have rooms in this demo. But what are in rooms in RE? For one, objects of interest. These objects were often pixel hunts in the older games; you walked along invisible edges of walkable space, trying to find what was usable against a bunch of backdrop materials. You do this here, too, but the game is working in your favor a bit more; objects that can be interacted with now have prompts, so you can quickly rule out what is important from what is just tonal detailing.
Items in the old RE games required inventory space. We have that here as well; in fact, our inventory seems even more limited! This follows from what we know about the character; a simpler individual probably isn't geared to hold much more than what we can in this demo. The clippers alone are large enough to take up two inventory slots.
We also see the player has "active" inventory slots. These are items which can be used immediately, regardless of context. The only item in the demo which does this is unsurprising in nature; it's a weapon. Using L2, the player can ready the weapon (which is also a defense stance) and using R2 use the weapon; using it against the dead carcass shows that the axe has been crafted to respond differently to different materials; this attention wouldn't likely be shown if we aren't expecting to need to chop up some zombies at a later point.
Movement is important to pay attention to. In a game primarily centered around stealth horror, where weapon usage is kept to a minimum, you'd expect to see a few things. First, you'd absolutely expect 'lean' functionality. Peaking around corners is paramount to a game like Amnesia where you are defenseless, and the same is true for Outlast. You'd also expect to see a "look behind" function -- used exhaustively in Outlast to determine how far away creatures and enemies are from you. Lastly, you'd expect to see hiding places; beds you can go under, closets you can climb in, etc.
Absolutely *none* of these elements are here. The only thing we can confirm is the controls and inventory management you'd expect out of a RE game. Nothing more, and nothing less.
In the original games, when you used a weapon (ready stance), you couldn't move. Here it seems as if you'll have some limited movement -- you can sprint, but I don't think we'll be running and gunning. This indicates to me that we'll get something nuanced; run, position yourself, and fire your weapon. Location damage will probably be important.
The limited inventory probably means we can expect to see some inventory management in the form of storage chests as in older games. The demo shows off some very basic puzzles of the form "get item->use item", but it looks like, since we have the option of discarding items (disabled in the demo), we'll need to smart about placement. I would suspect that larger puzzles will show up.
Of course, the game will take place in a mansion area, just as RE1 did. This is about as on-the-nose as you can be with regards to RE DNA. Everything they've built here seems to be a wink and a nod to RE1 when you use a bit of your imagination to see how these systems will work in a much larger game. Unlike PT, the demo here is a more "full-bodied" experience; it has deeper exploration, and in those moments of exploration you can start to get a sense of how interesting this kind of gameplay can fit in an RE mold. Yes, it's different from RE6, but it's not actually much different from the slower pace of RE1.
I can't argue that it appears, tonally, that they are going for something much darker and less camp. If those are the elements of RE you love most, I'd brace for impact.
There is no indication in this demo of any kind of player health, etc. Mechanically, we're missing some things that will obviously be there, unless death is going to be "instant" on hit -- which is a possibility, but I'd imagine unlikely.
My conclusion is that this demo is not very RE-like, but does support everything we'd expect from RE1, if RE1 were in first-person. Movement will still be important. Shooting will be, too. Melee weapons will probably play a much larger role. Exploration will be a key part of the game, at a slower pace. It will be darker, and have more horror influence, but it will *not* be like Amnesia, or Outlast, and the PT elements found in this demo will likely *not* find their way in the full game (if they did, nobody would ever be able to finish it

).
So fear not RE fans, and wait for a reveal of gameplay from the actual game. In the mean time, try the demo; looking at it from a different perspective (pure mechanics, fluidity, exploration) may actually impress you. The full game will likely introduce the characters you enjoy, and a scary room that turns you into a Jill sandwich.