Verschiedene Zugänge (Gewerbe und Privat) und verschiedene Steuern anzeigen,

Hi,

When you configure tax rules in the System>Taxes menu, you have a "client type" where you can choose between "individual" and "company without a VAT number". Then, you can enable the "address_company" setting of the Display>Custom fields menu.
That way, when people register/enter their address, they can enter a name if they are a company or leave that field empty if not.
And then, based on that, the tax rules will apply to them or not when they are logged in.

This however means that for people that see the prices without tax, they won't pay taxes.

If you goal is only to display the prices without tax on the products listing when the customer is a company and still charge the taxes on the checkout, then this won't work out for you.

What you can do is configure your tax rules for everyone, and then have two menu items configured the same way to access the shop, one for the individuals, and one for the companies.
Then, in the "products options" tab of the menu items, you can configure the "Show taxed prices" setting differently so that the prices will display with taxes for the menu item for individuals and without taxes for the menu item for companies.
Note that this is only for the listing.

Now, you could also create two different websites. That's what most companies do so that besides the taxed prices, they can display different things for individuals and companies. For example, doing that, you showcase products dedicates to companies on the companies website, and products dedicates to individuals on the individuals website.