No, there's no certificate, but there is a process of documentation and testing commonly referred to as "certification" before you are allowed to process credit card transactions.
This depends entirely on the organization and their acquiring bank's requirements (ultimately the acquiring bank is the only one who matters, but most reasonably organizations develop their own process to ensure they're covered as much as possible). For many small businesses, they're often times just buying a cheap terminal and swiping away. The acquiring bank isn't pressing them for details of their security measures and they're often completely clueless about any requirements they're supposed to be meeting. They aren't bringing in a QSA. Even if they were, bring in three QSAs to any decently sized organization and get three different opinions about your scope and your compliance measures. Half the fun of PCI assessments is determining what the requirements mean, how they apply in your specific instance, and where scope ends. But the point is, there's no issuing authority to say that you're PCI compliant. There's no governing body certifying anyone. The only thing that's actually there are the contractual relationships between the merchant and the acquiring bank and the contractual relationships between the acquiring bank and the payment brands.
I work in point of sale software development and have had to help retail chains overcome problems found in their certification tests. You either don't know what you're talking about, or you're playing a pointless semantic game.
It's not a pointless semantic game because it's the unspoken risk for anyone accepting credit cards. Since there is no official PCI certification and since there is no agreement between QSAs on what the requirements mean in principle (let alone in practice in a specific organization's situation), the PCI SSC gets to stick the claim up on their website that no breach has ever occurred in a PCI-compliant vendor. Best of all, each individual payment brand actually gets to decide what requirements have to be met in which situation by which type of vendor doing what type of business at what scale and via which medium. The ambiguity and the leverage the payment brands hold allows them to arbitrarily decide who is and who isn't compliant at any given moment.
So you keep on doing your documentation and your testing processes (and you should, it's good practice), but if you think for a second your customers are somehow protected from Visa, Mastercard, etc in the event of a breach, you'd best think again. It's a shell game designed to ensure that whenever things go south, the payment brands are never the ones left holding the bag.