It's apparently a very British thing to love an underdog, just as it's very American to love winners and get drunk on weak lager and shout about it. Why that is I couldn't say, not that that's going to stop me from trying. It might be because the British are so used to winning things, like wars and
Perhaps we prefer underdogs because this country is so class-ridden that most of us are underdogs from the moment we're born (the radical's perspective) or perhaps it's that we're just bitterly jealous of anyone who sullies themselves enough to make it in the world (I don't know what perspective that would be). I don't have any clear idea of what it is we see in underdogs; perhaps we admire their pluck, their courage to keep going when they're clearly going to lose - Eddie the Eagle springs to mind - their courage to be known for being a total loser their entire life. Having said that, Eddie might say that in some way he won, because he won the hearts of a million British people who were like 'fair play to him having a go at something like that, I'd shit myself', and also to be fair he was the best ski jumper in Britain by a fair distance, so not a loser at all in fact, when you come to think about it.
I don't think we admire pluck, I think we like underdogs because they don't make us feel bad or at least any worse about our miserable lives, which is why, as soon as they stop being underdogs and start being overdogs, or whatever they become, hot dogs maybe, we drop them like a hot potato and cast around for someone else more underdoggy.
Dropped like a hot potato's a funny term isn't it, I mean when was the last time you picked up a hot potato? Not, of course, that it's not possible to pick up a hot potato, I'm sure that it is, I'm sure in olden days sailors used to pick up hot potatoes on the docks all the time, or something, and possibly caught hot potatoes in the process.
So where does the term underdog come from? It could be from greyhounds, but they rarely seem to be under, more to the side or just behind. Could it be that Britain's favourite animal metaphor is to do with the receptive dog in coitus? Perhaps the British only like people who are getting fucked by the top dog, but then how does that fit in with the traditional idea of the underdog being someone who comes from behind? What else could it be, underdog, a dog that used to go underground that everyone liked? Maybe it's related to dogging, that favourite British past-time, maybe Britain loves underdogging, no, I don't think I'm going to pursue that line of thinking.
Of course underdog could come from dog fighting, as in the dog that submits (under) the top dog, but in that case the underdog would be the loser, whereas underdog tends to mean the least fancied before the event, the David going up against the Goliath, and in fact there's an implied meaning of underdog that they often go on to win, otherwise they're just losers.
So I don't have a clue. A bit of research did turn up this attempt at etymology, but apart from being extremely tenuous, it's totally unreferenced and seems to be, more to the point, nonsense.