Is the moon phase the same across the globe?

Yes — the illuminated fraction is identical everywhere on Earth at any given moment. The moon is ~384,000 km from us; Earth is only ~13,000 km across. From the moon's perspective everyone on the daylight-side of the planet sees essentially the same lit hemisphere.

What does change is the orientation and the local time of moonrise/moonset:

• In the northern hemisphere a waxing crescent looks like a "D" (lit on the right). In the southern hemisphere the same crescent looks like a backwards "C" (lit on the left). At the equator it's lit on top or bottom.
• Moonrise can be hours apart between, say, Tokyo and New York for the same calendar day — but the phase you see is the same, just at different points in the local night.

Why does the moon change shape?

The moon doesn't actually change shape — only the part lit by the sun does. As the moon orbits Earth (once every 29.53 days, the synodic month), the angle between the sun, moon, and Earth changes. We see different fractions of the moon's lit hemisphere: a thin sliver near new moon, a fully lit disc at full moon, and everything in between.

Why is full moon "full"?

At full moon, Earth sits roughly between the sun and the moon, so the side of the moon facing us is fully illuminated. Because the orbits aren't exactly aligned (the moon's path is tilted ~5° relative to Earth's), most full moons skip the Earth's shadow — but when they don't, you get a lunar eclipse.

Waxing vs. waning

Waxing means the lit portion is growing (new → full). Waning means it's shrinking (full → new). A handy mnemonic in the northern hemisphere: a "D"-shaped moon is waxing toward full, a "C"-shaped moon is waning toward new. (It's flipped in the southern hemisphere.)

Why isn't there an eclipse every full moon?

The moon's orbit is tilted about 5° relative to Earth's orbit around the sun. Most full moons pass slightly above or below Earth's shadow. Two to four times a year the geometry lines up and we get a lunar eclipse.