1. A possibility is that, while we can't define things, we recognise them when we see them (e.g. Platonic Ideals: point, line, circle, plane). However, this makes it a religious matter. If two people don't recognise the same thing we have no way to resolve the issue. No matter how many people argue that their view of God is the right one, we don't even know exactly what it is they are arguing in favour of.
It isn't that one group is right or wrong - it is that we can't make any progress. If somebody comes along and says "The Empty Set is this completely other thing" then mathematicians can say the person is wrong - but they can't demonstrate it.
2. The way that "The Empty Set" actually works is through the relationships we have with that phrase. The actual meaning of the phrase is utterly irrelevant - more - it doesn't have a meaning - The only thing the phrase "The Empty Set" represents is the set of relationships. When you "know it when I see it" - you aren't recognising a 'thing' - you are recognising the shape of a group of patterns. When working with "The Empty Set" we are, in fact, simply manipulating relationships.
3. Maybe there really are platonic ideals. But they are completely irrelevant to us. A thing that we cannot describe, whose properties we cannot articulate, measure or perceive in any way simply doesn't matter.