TYPO3 Core

If using the environment variable the l10n cache in .build/var/cache/data/l10n does not contain the translation (Screenshot 1).

If the env variable is not set, the cache is under var/cache/ and the l10 cache includes the german translation. (Screenshot 2).

Instead of setting the `TYPO3_PATH_APP` environment variable directly,
have you tried to use following in your extension composer.json to
setup the TYPO3 installation ?

{
    "config": {
        "allow-plugins": {
            "typo3/class-alias-loader": true,
            "typo3/cms-composer-installers": true
        },
        "bin-dir": ".Build/bin",
        "vendor-dir": ".Build/vendor",
    },
    "extra": {
        "typo3/cms": {
            "extension-key": "your_extension_key",
            "web-dir": ".Build/Web" 
        }
    },  
}

That will create a full TYPO3 composer based installation in `.Build/` with `./Build/Web` as "public" folder (document root for ddev for the domain)
and properly bootstrap all path calculations properly ?

At least, this is what we normally use for extensions development (and I'm settupt up my local apache2 on ./Build/Web and never detected an issue.

I'd would stay away of manipulating these environment variables by hand as there are multiple involved and can lead to undesired effects.