Okay. I'll explain it like this.

Lich Banes Passive is granting addional attack damage after using a Skill. Yes this item works on other mages as well, but it's not the point. The point is that Sona works very very well with this item, because Sona also works well with cooldown reduction causing her to have very low cooldowns. Let me mention here, that even without additional cdr her cds are not that high.
Sonas Trades don't end with either her Q or her powerchord, because she can still do additional damage due to Lich Bane on almost every AA, if you time it right.
And honestly? How much damage does Diana do with her Q at lvl 1? We are not talking about what happens at level 1, because usually nothing happens at that level. Everyone tries to out-cs their lane-opponent so they get to lvl 2 first. Also no one has a Lich Bane, or even a Sheen at lvl 1 (Remember ARAM starts at lv3).
Sona often reaches the point in game, where she can continually spam her q w e without much of a brake in it. So she goes: q, aa, w, aa, e, aa, q, aa .... and so on, once in a while triggering her Ult. In Addition to that you might get a Powerchord, that is empowered by using Q before and the Lichbane. And this does a lot of damage. (as does her q in generall)
Did you never wonder why people kept banning Sona in URF? It's not only because of her heal, I can tell you that much. Other champions can heal as well...
Another strength of Sonas by the way is her low Skill-Level. She has no skillshots (Her Ult is so big that is qualifies as a Skill-Wave but not a "shot") and she doesn't have any point and klick skills. Only Skill-Activation and Auto-Attack. This makes her especially effective, because it doesn't allow for dodges.
By the way Diana, TF and Fizz build Lich Bane because they have to AA due to their skills/passives anyways. The Item does well on them too, but for different reasons.
To be honest the best comparison of Sona -Lich Bane synergy that comes to my mind is Irelia - Trinity Force.