You might be confusing two things:

1. The active prefix in Tailwind CSS is applied, whenever an element is currently pressed. (e.g. while you press on a link or button) (https://tailwindcss.com/docs/hover-focus-and-other-states#active)
2. The currently "active" page, as in the current page path matches the nav item`s href

You cannot achieve 2 with 1, they are different things.

To achieve 2 in next.js, you would need to get the current path, compare it to the <Link> element's path and if they're equal, add conditional tailwind classes that show that you're currently on the respective page.

Here is a code example how you can achieve 2:

import Link from 'next/link';
import { useRouter } from 'next/router';

export const Header = () => {
  const router = useRouter();

  return (
    <header>
      <Link href="/">
        <a className={router.pathname == "/" ? "active" : ""}>
           Home
        </a>
      </Link>
    </header>
  )
}

Source: https://dev.to/yuridevat/how-to-add-styling-to-an-active-link-in-nextjs-593e

In your case, you could do something like this:

<Link href="/user/signup">
    <a className={`mr-4 my-auto hover:text-indigo-600 font-normal font-serif text-xl ${router.pathname == "/user/signup" ? "text-indigo-600" : "text-brand-darkblue"}`}>
        Home
    </a>
</Link>

You need to use NavLink instead of Link it will match the current URL here is an example I've used :

<NavLink to={`/${link.name}`}  className={ ({isActive})=>  (" capitalize my-2 flex flex-col ") + (isActive?ActiveLink:NormalLink) }>
              {link.name}
  </NavLink>

Using NextJs and Tailwind using toggle dark/light mode, this is how I solved it

    //ActiveLink.js
        import { useRouter } from 'next/router';
        import Link from 'next/link';
        import PropTypes from 'prop-types';
        import { useTheme } from 'next-themes';
        
        
        const ActiveLink = ({ href, children, ...rest }) => {
            const router = useRouter();
            const { theme, systemTheme } = useTheme();
            const useLoaded = () => {
               const [loaded, setLoaded] = useState(false);
               useEffect(() => setLoaded(true), []);
               return loaded;
            };
            const mounted = useLoaded();
        
            const isActive = router.asPath === href;
    const currentTheme =
            mounted && theme !== undefined && theme === 'system' ? systemTheme : theme;
        
            const activeLinkBgColor =
                currentTheme === 'dark'
                    ? 'bg-gray-700 text-white'
                    : 'bg-blue-600 text-white';
        
            const themeBgHover =
                currentTheme === 'dark'
                    ? 'hover:bg-gray-700 hover:text-white'
                    : 'hover:bg-blue-600 hover:text-white ';
        
            const activeLinkAndNotActiveColor = isActive
                ? activeLinkBgColor
                : `text-gray-300 ${themeBgHover}`;
        
            const className = `${activeLinkAndNotActiveColor} px-3 py-2 rounded-md text-sm font-medium`;
        
            return (
                <Link href={href} passHref {...rest}>
                    <a className={className} aria-current={isActive ? 'page' : undefined}>
                        {children}
                    </a>
                </Link>
            );
        };
        
        ActiveLink.propTypes = {
            href: PropTypes.string.isRequired,
        };   
        
        
        export default ActiveLink;

// How to use it
 <ActiveLink href='/about'> About </ActiveLink>

Result:

Good luck.