How does my VAT ID affect the price?

On a Shopify bill, tax is Shopify's to apply. Shopify adds whatever tax it must charge in your region. Any VAT details for that bill are kept in your Shopify account, not in nouz.

If nouz bills you directly in euro, your VAT ID can change the invoice. A VAT ID is the tax number of a business in the EU. Say your business is in Germany, and you enter a valid VAT ID. Then no VAT is added, and your own tax return accounts for it. This is called reverse charge.

In Austria, where nouz is based, VAT is always added, whatever ID you hold. Outside the EU, no EU VAT applies. The plan prices nouz shows are net, which means before tax. Stripe Tax works out the tax at checkout. That is why the total changes as soon as you enter your country and your VAT ID.