It works almost everywhere, except some “special” clients, so in case someone needs this in the future, here is the solution https://shopify.dev/tutorials/authenticate-a-private-app-with-shopify-admin#:~:text=Basic%20authentication,to%20connect%20with%20your%20app.
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Join the API key and password with a single colon (:).
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Encode the resulting string in base64 representation.
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Prepend the base64-encoded string with Basic and a spaceadd this as header: Authorization: Basic NDQ3OGViN2FjMTM4YTEzNjg1MmJhYmQ4NjE5NTZjMTk6M2U1YTZlZGVjNzFlYWIwMzk0MjJjNjQ0NGQwMjY1OWQ=