I'm using nuxtjs with tailwind module and have same problem, following steps work for me.

1. Change tailwind.config.js to .ts, and replace @type {import('tailwindcss').Config} with satisfies Config like this.

import type { Config } from 'tailwindcss';
import colors from 'tailwindcss/colors';

export default {
  content: [],
  theme: {
    extend: {
      colors: {
        primary: { ...colors.sky, DEFAULT: colors.sky['500'] }
        //...
      }
    }
  },
  plugins: []
} satisfies Config;

2. Other ts file that use config

import resolveConfig from 'tailwindcss/resolveConfig';
import tailwindConfigRaw from '@/tailwind.config';
const tailwindConfig = resolveConfig(tailwindConfigRaw);
tailwindConfig.theme.colors.primary.DEFAULT // hover will show: (property) DEFAULT: "#0ea5e9" 

hope this help :D