Real values, drop-in shape.
Suppose the approved brand color is #6366F1. Tailwind's own indigo scale is anchored on that exact hex, which makes it an honest worked example. The extend block looks like this: brand: { 50: "#EEF2FF", 100: "#E0E7FF", 300: "#A5B4FC", 500: "#6366F1", 600: "#4F46E5", 700: "#4338CA", 900: "#312E81" }, and components immediately gain bg-brand-50 card washes, bg-brand-500 buttons, and hover:bg-brand-600 states.
Notice the shape of that scale: 50 is a near-white wash, the jump from 500 to 600 is big enough to register as a hover but small enough to stay on-brand, and 900 is dark enough for text on the 50. When your brand color isn't a stock Tailwind hue, the generator at the top of the Tailwind Color Palette page builds the same 50 to 950 shape from your hex, following the lightness curve of the default palette in OKLCH, and exports it as a v3 config object or a v4 @theme block.
Before generating anything, paste your brand hex into the Tailwind Color Palette's finder. It lists the three closest default colors with a Delta E score each. If the nearest one is very close, using the stock scale may be simpler than maintaining a custom one. The hex values shown are Tailwind v3's; v4 defines its default palette in OKLCH, so v4 colors can look slightly different.