When people describe you or your company what is the first thing they say? What do they lead with? That's your leading characteristic (at least in that particular context).
He's the _______ guy. That's the _______ startup. Isn't that the __________ search engine?
Unfortunately, the way we are wired means we generally don't like to put more than one thing in those blanks even though most people and companies would prefer more words. It's mission statement vs mantra.
Another unfortunate thing is that it is often a discriminating characteristic, more or less so depending on the situation. For example, suppose you went to a party with a friend and were trying to relate later to them someone you both saw from afar. You have an incentive (in terms of shortening the exchange) to name some physical feature that made them stand out in the crowd, e.g. height, skin color, weight, handicap, speech pattern, clothing type, etc.
Most people would have a strong negative emotional reaction to overhearing that conversation about themselves because they don't consider that feature their overall leading characteristic in terms of identity. And yet it is makes sense to use it in that context.
Take the slightly different context of telling someone about someone else who they've never met and will not imminently meet. In this context, physical features matter less but people often still lead with them. Or even worse they lead with an affiliation like religion or nationality that has little or no discriminating value unless both of you have the same stereotype of that group and the person you're talking about matches it exactly.
I caught myself doing this the other day to my dismay ("he's Canadian"). And I distinctly remember a long drawn out argument on this subject after my friend in high-school did the same ("she's Jewish"). It's insidious. The behavior furthers stereotypes without them being validated as accurate and the words themselves often don't convey much information at all.
In other words, people often make poor choices of leading characteristics. They take the path of least resistance, insert their own biases, repeat hearsay, etc.
Startups have the same problem, but they can use it to their advantage. What people use as the leading characteristic for startups usually makes good marketing material because it aligns with peoples' natural tendencies. At the very least, you should know how people are relating you to others.
The next step is to present your product in different ways and see if different leading characteristics fall out. Sometimes you can frame your product with initial marketing, and in other cases you will tailor it in certain ways to get the desired effect.
In any case, the end result can be dramatic in how well your product spreads by word of mouth. Does that leading characteristic resonate positively with people or not? The same is true on a personal level, but unfortunately we have less control over that one.