What we call “a lunar eclipse” is a solar eclipse in the Moon. What we call a solar eclipse is a solar eclipse on Earth. We can also observe solar eclipses (total or partial) on other bodies in the solar system.
I agree that both are real, with a real shadow. But for a (total) lunar eclipse the full Moon is in the shadows, while in a solar eclipse a little shadow traverses the Earth surface.
In case of lunar eclipse, the view of the Moon is obscured by the shadow of Earth. I.e. Sun's light is blocked by Earth, so it doesn't reflect back to Earth from the Moon. In case of solar eclipse, the view of the Sun is obscured by the Moon itself (which throws shadow on Earth), i.e. you don't see direct light from the Sun blocked by the Moon.
Both are pretty real and both are called eclipses.
And Sun still shines the same during lunar eclipse too. Only those under the shadow of the Earth see the difference. It is just that Earth's shadow is big enough to cover the Moon and that's it. Neither eclipse is more or less "real" than the other one.