While it's true that any character of an appropriate age will have been somehow affected by the Last War in Eberron (assuming the DM hasn't removed it), that's not really limiting.
Firstly, you only need to decide what happened to him if and when it comes up. And you always have the option of saying, "I don't want to talk about it." Or even have your character suffer amnesia.
Secondly, just because someone was affected by the war doesn't mean he has a depressing tale to tell. Perhaps he found his purpose in life in the war. Perhaps he met his one true love. Perhaps he was lucky, and everyone he cared about got out safe. Or, of course, perhaps he spent the war far away from the conflict.
The truth is, if I were running Eberron I would run a different campaign than I would in Forgotten Realms, even if I ran all the same adventures and had the same mix of PC races and classes. But that's not because either campaign is limited, it's because they have distinctly different flavours. And that's a good thing.
Of course, if you don't like the flavour of Eberron, or Forgotten Realms, or any other setting of your choice, that's cool too.