It is a business. They're selling a service. I don't know why they're protesting at the notion of being a company, they're de facto a business (selling service behind a brand, which they're openly promoting to sell more services).
(WebP Cloud Services starts by providing a free service of Gravatar/GitHub Avatar reverse proxy with WebP optimization at first, and now it's our first attempt to make a paid services of private proxy as more of our users want this to be a more generally available service.
(And we are currently not a company indeed) ´･ᴗ･`
No intentional protesting at the notion of being a company, just unsure if "company," "business," and "startup" have the same meaning in certain contexts.
Do you have any advice in this regard? We do have a preliminary plan to register a European company in Estonia (through e-Estonia) after achieving good revenue to continue our operations.
You need a company to own things, such as the IP (code, trademarks, website, customer lists), as well as being the thing to which revenue is paid. You'll also find you can't do most things without it (such as get a credit card, office lease, cloud discounts, etc).
Most importantly, suppose you have a cofounder break up when you have just started getting "good revenue" but haven't yet got a company. Who's is that revenue? Who owns the code you wrote? A complete mess.
I don't know anything about e-Estonia, but if they allow you to sign up today, no reason not to do that. In the US (or abroad if you want a US company), Stripe Atlas is a good option. That might work for you too.