Tuning changes with wood density. For example, heavy mahogany body guitars tend to stay in tune much better than lighter weight bodies. It's something with the wood density, temperature, and changes in humidity. As the guitar goes slightly out of tune, the guitarist changes finger pressure on the strings which certainly impacts tone.
The strings also vibrate differently, affecting the field around the pickup. You could put pickups on a pine or balsa wood body and the guitar will definitely sound different than on a denser wood that doesn't lessen the vibration of the strings. A main goal of a solid-body guitar is to remove resonance to lessen feedback and the stiffer and denser the wood, the less the feedback.
Wood density also affect tones by the amount of stray electrical signals the wood can block. Shielding the pickups and wiring lessens the difference between different types of wood but back in the day, shielding guitars was a rarity. Some of the old guitars with their odd woods, coupled with cloth insulated wires and lack of shielding had a distinct tone that some guitarists favor and will pay top dollar for. The paint or varnish also affects how the stray electrical signals are blocked which is why some guitarists want the raw wood exposed.
Wood also affects how users hold the guitar. Neck dives on Gibson / Epiphone SGs are a pain. Not all of them have it and not all dives are as extreme but a heavier body and a lighter neck would allow a guitarist to hold a more relaxed grip on the neck which would affect tone. The alternative of moving the strap from the front button and tying it to the headstock changes the tone. Yes it's the strap location putting tension on the neck affecting tone, but it's the wood that led to the strap being reloated.
Here's a picture of Sister Loretta Tharpe playing her Gibson SG with the strap tied to the headstock instead of buttoned to front of the body:
For an idea of how the tension the neck sounds, here's a video of her playing "Trouble In Mind" on her Gibson SG:
https://www.youtube.com/watch?v=U3PNc_cWv9M
There's a punchiness to it, a lessened sustain to the notes, even more than Gibsons with their shorter scale have.