A "35mm camera" is one that uses 35mm-wide film stock. The most common stock fits in 135 cartridges, but that's not a requirement -- note that the first Leica used 35mm rollfilm, and some setups (including for old Nikon, Alpa and Miranda SLRs) used 100-foot rolls of 35mm film.
What I call 135/FF uses 35mm film in 135 carts with 36x24mm frames. That's not the only 135 frame size. I've had and still have 135/HF (half-frame) still cameras, both RFs and SLRs. Ah, my long-lost Olympus Pen-FT... and Canon Dial-35... Others, notably the Robot cameras, shot 24x24mm square frames. Some 35mm panorama cameras shoot P1 (58x24mm) or P2 (65x24mm) frames, or even longer for 360-degree panos. Also note that 126 film carts are loaded with 35mm-wide film stock, but in a different package with 26x26mm frames. 126 cameras *can* be called 35mm. Go figure.
So, the first answer: "35mm camera" means it shoots 35mm film; and commonly (but not always) means 36x24mm frames. Exceptions abound.
Sensor sizes: Canon uses 36x24mm sensors. Nikon's 'FX' sensors are 36x23.9mm. I don't have the Sony numbers in front of me; but since Nikon buys sensors from Sony, I'll assume that Sony FF is also 36x23.9mm. So they *do* replicate the format -- but not the resolution. IIRC a high-res scan of a fine-grain 135/FF slide produces something like a 100mpx file.
Also note that while the term FF often appears in a 135 context, it has other uses:
- The old standard cine frame was 24x18mm, what we'd call Half-Frame but what cine users called Full-Frame. In cine, 36x24mm is Double-Frame.
. - The 645 film format has 56x41.5mm frames, not quite 60x45mm. The Pentax 645D produces 44x33mm frames. There's talk of 645/FF digital sensors. So in this context, FF means "a sensor the same size as a film frame".
. - The nominal (official) APS-C frame size is 25.1x16.7mm, with a diagonal of 30.1mm. My K20D's so-called APS-C sensor is 23.4x15.6mm, with a diagonal of 28.1mm. If an APS sensor actually was the nominal size, would it be APS-C/FF?