Jeez, no wonder people think that combat is trash in this game - even the seemingly simple things are so hard to understand sometimes.
Chaining attacks does increase damage per hit. The increase in damage comes from filling up the combo bar (bottom left).
Filling up the combo bar beyond the "special attack" threshold does add a noticeable damage bonus per hit, as well as attack strength (chance to stagger enemy on hit).
You will need to learn to time your swings, of course, in order to fill up the bar as fast as possible. Mindless button mashing will get you nowhere, as per original G1+2 combat design.
However, the flat damage reduction threshold of enemy still has to be overcome in order for that damage increase to be visible in the first place.
Needless to say, it's retarded to expect to see any good results vs a skull-grade enemy while bashing them with an iron bar; you still need a good weapon that can penetrate DR in the first place.
When doing long attack chains (5+ successful consecutive hits), I've definitely seen an increase in damage per hit - IF I timed the swings correctly to fill up the combo bar + IF I was fighting against gear-appropriate enemy. It was easily noticeable because I was using a non-enchanted (no DoT) weapon at the time.
EDIT: play-tested it to prove my words. Results - negative.
Went in vs Ripper (a lizard, 60 exp) on Difficult.
Used a zerker 2-hander + (55 dmg), no enchantment, melee I skill (+10%).
A successfully-executed combo of 5 consecutive light hits WITH filled-up combo bar (filled to 60% approx) took it down to 1/3 of its health.
Then I did the same, but with button-mashing. Again, a successful 5-hit combo with an almost empty combo bar got it down to exactly same amount of health (I took screenshots and compared the two life bars).
For a final test, I hit that ripper 5 times, but not in a combo - a hit, then empty the combo bar, then hit again. I took it down to merely 2/3 of its health.
For me, that means two things:
1) Successfully chaining several hits in a combo does significantly increase the damage dealt per consecutive hit.
2) Filling up the combo bar does not add any damage on top of that. Whether it still adds attack strength (aka stun chance) as it fills up - I don't know, but I will still assume that it does.
Corollaries:
1) You can get away with some degree of button mashing with a 2h weapon since it allows to stun-lock most opponents while you perform a combo. Risks: you can't easily stop a button-mashed combo, meaning you WILL get fucked vs multiple enemies or enemies with high parry strength if you button mash. 1h weapons can't stun as well, so you definitely need to be more careful in order to pull off those multi-hit combos.
2) If you want to deal good damage in melee, you pretty much have to learn how to chain-hit enemies.
P.S.: I suppose I could also test the influence of the combo bar on the attack strength for 1h weapons, since 2h weapons in general can stagger most enemies from the 1st hit onward. Plus, I can also try to check how much damage consecutive hits in a combo add. I have an unenchanted 1h regent sword ++ (64 dmg), so it should be able to inflict noticeably more raw damage per hit vs dem ripper lizards than a 55 dmg 2-hander. Will check it once I get back from work.