More of a legal history perspective, but contracts historically were based on trust. Basically you could cash a check because you said you were the person on the account. You would only need to prove your identity if there was something wrong with the transaction (someone else claimed it, the check was returned, etc.).

If in court you needed to prove your identity, you would have to do so by a preponderance of the evidence, which is to say that you would prove that it is more likely than not that you are who you say you are. You would use whatever you had at this point, family bible, a deed, a birth certificate if you had one, etc.

You might be interested in The Return of Martin Guerre by Natalie Davis. It's a historical analysis/reconstruction of a court case in 16th century France that deals explicitly with the subject of identity. A basic synopsis of the case is that after a man named Martin Guerre left his family, an imposter came and took his identity. After people started to figure him out there was a trial to determine his identity, and as if the whole situation wasn't dramatic enough, the true Martin Guerre returned in the middle of the trial. Davis' analysis reconstructs both the life of Martin Guerre and the trial itself using archival evidence and the papers of the judge who presided over the case and led the investigation. It's been awhile since I've read it so I'm murky on the specifics, nevertheless it should give an interesting perspective on your question.

Missing from the comments below are notes, and the letter of credit. If you are a traveler, going from London to Madrid, you would go to a business or bank or even a friend in London who had a contact, branch or associate in Madrid. They could take your money, and provide you a letter for their associate saying, in essence "this is George, give him 40 reals". You would have some other documents, even letters of introduction, passes from the Spanish embassy, with you, showing your name was George. There would usually be a discount- i.e., they would give you a little less than your 40 reals, to make a profit. This is how you'd avoid hauling bags of gold with you.

But for more local transactions, especially with a scarcity of small money, there would be "notes". You build a doghouse for George Washington, he pays you with a note saying reimburse the bearer 2 shillings. You can then pay somebody else with that note, when you owe them. Eventually somebody will show up at Washington's house wanting the real cash, and he'd pay them...or tell them to come back Monday.

The important thing is, notes often did not require ID, anymore than money does today. BUT if somebody gave you a note, or letter of credit, and you doubted it was good, you could decline to accept it, or you could discount it heavily- give them much less than the face value. So, your note from Washington would be less discounted than a note from somebody totally unknown. If the note was for a specific person, you; you could sign it over to someone else, and they could then sign it over to someone else; and anyone could also refuse the note, or discount it.

A check was essentially also a note, but instead of eventually presenting it to Washington, someone had to present it to the bank. And the bank itself could simply issue notes- which is where you get paper money being called bank notes.
From here you can see how notes would be used as credit, loans, be traded, used as collateral...and it gets complicated.

While there is always more that can be added, so I don't wish to discourage anyone from answering, you'll find that there are older threads which might interest you as a starting place at least. Check 'em out here!