The answer to the question is in the word “pedigree”. The word pedigree is defined as:
“the record of descent of an animal, showing it to be pure-bred.”
As you can see you need a record of the cat’s family tree in a document. The family tree will show that the ancestors of the cat are the same breed. If they are not the same breed then the cats will be random bred cats or cats of another breed allowed by a cat association such as the Cat Fanciers’ Assocation in the USA.
Lastly, the cat’s pedigree needs to be certified (accepted and agreed) by a recognised cat association or cat registry. That is how you know if your cat is a pedigree.
People sometimes ask if their cat is “purebred”. They are asking the same question effectively. The concept of purebred means that cats of the same breed are in the family history and therefore the genetics of the cat are “pure”, meaning untainted by unacceptable (to a cat association) genes of another breed or random bred cat(s).
You will only know for sure if your cat is pedigree if you have some sort of formal documentation (paper or digital) to prove it.
There are a lot of cats in America who look like purebred, pedigree cats. They may be pedigree cats. They often look very beautiful. Althoug they look like pedigree cats but you can’t be sure until and unless you have the paperwork to prove it.
This is why a lot of people leave comments on this website asking whether their cat belongs to a certain breed. Often times the cat does look as if he/she belongs to a breed because of the outstanding appearance. However, nobody can be sure until you have the family tree and registration with a cat registry. Under the circumstances, you have to describe the cat as a purebred-mix or, for example, Siamese-mix or even “domestic shorthair” if the cat is a shorthaired cat.
It appears that sometimes when a cat breeder closes her hobby business she places her cats with a rescue organisation whereupon a lucky person adopts one of the cats without the necessary documentation to prove that the cat is a pedigree cat. Then they ask the question: “How do I know if my cat is pedigree?”
Appearance leads you to believing your cat is a pedigree cat but you’ll need more.
P.S. The difference between “purebred” and “pedigree” is that the pedigree shows the family history of the cat leading to the conclusion that the cat is purebred.