In the US there are ~0 rules about invoices. I'm in the middle of implementing EU-compliant invoices for a US-based SaaS offering and boy, y'all need to chill about all this.
Launching the actual product was easy. Complying with EU rules has been the nightmare.
VAT (Value Added Tax) is the main reason why EU is more strict about invoices than US. And these rules are there for a good reason given many different ways how VAT system can be exploited.
What are those reasons? (Seriously asking, not being snarky)
The US has sales tax but doesn't require any of these rules around invoicing.
It mainly revolves around the facts that 1) VAT is deductible for companies in certain cases 2) Companies are responsible for collecting the VAT for products they sell and submitting the collected money to the government. These things mean that auditors (and in some cases officials) must be able to verify that VAT has been collected, paid and deducted correctly. Therefore it is important to have receipts which clearly show who sold, who purchased, what was purchased, what was the price and how large portion of price was VAT.
Not sure how it's in US, but in those few European countries I'm familiar with you need a proper receipt (showing what was bought, VAT etc) for bookkeeping. Credit card statement is not enough.