The EU VAT Assistant doesn’t calculate, nor deduct VAT from the prices. When a valid VAT number is entered, the plugin tells WooCommerce that the customer should be exempt from tax, then WooCommerce (and, possibly, other plugins), do what they need with that information.
In normal circumstances, the result is that the whole order is made exempt of VAT, not just products. If the shipping still shows tax, then it means that “something” is applying the tax to that element. For example:
1. A shipping plugin, which fetches costs and taxes from a remote service, ignoring the exemption.
2. Another plugin (not necessarily for shipping) that adds taxes to the shipping cost.
3. Some custom code in the theme.
4. some automated tax calculation service that ignores the exemption and applies taxes anyway.
The first step would be to identify such element. A simple approach is to go by exclusion, disabling other plugins one by one and switching to a basic theme to exclude custom code. That helps narrowing down the list of possible causes, making it easier to identify the source of the “extra” VAT on shipping.
If you got a support plan for the EU VAT Assistant, we can also investigate the issue for you (such service is not available as part of the free support provided via this forum). In such case, you can contact us via our contact form, at https://aelia.co/contact, and we will send you further instructions.