There is a real material called rainbow obsidian, and it comes principally from Jalisco in Mexico. Its colour is not body colour: it is a sheen produced by aligned microscopic inclusions, and it appears as concentric or parallel bands of bronze, green, violet and gold that show only when the stone is cut and oriented correctly and viewed at the right angle. Turn the piece and the bands vanish, leaving plain black glass.
That behaviour is the whole test. If the colour is present from every angle and in every light, it is body colour, and obsidian does not have body colour in those hues. Bright, permanently coloured 'rainbow obsidian' beads are slag glass.
Cutting genuine material well is skilled work, because the sheen bands lie in a specific plane and a cabochon has to be oriented to them. That is why good examples cost more than the raw material suggests, and why cheap ones are usually not what they say.