Are checks from the bank are unique? Can you give a blank check to a friend so that they can use it as if it were their own? Or must checks be linked to a person's bank account?
The bottom of the check will have the bank's routing number, the account number, and the check number—these uniquely identify that specific check to your account.
This will show at the bottom of the check like: 1234567890 555444333321 00001
Routing number: 1234567890
Account number: 555444333321
Check number: 1
In the "distant" past, checks were in fact blank, except for the bank name and (in the US) ABA routing number.
In the 1950s, though, banks -- and the companies that make check processing equipment -- standardized on fonts and magnetic ink, which allowed the equipment to process checks faster.
This, though, necessitated that the routing number, person's account number and the check number be printed on the check in that magnetic ink.
Here's one from 1935 with the account owner's name printed on the end of the check:
And here's one from 1908 which is completely generic except for the bank name:
Just to add a bit of fun color to this, the answer is that the routing number, account number and check number uniquely define any check. However, this should come with a caveat.
After the 9/11 terrorist attack in New York and the Pentagon, the FAA grounded all aircraft in US airspace for several days. Something that is not widely known is that during that time the American banking system ground to a halt. The reason being that at the time checks were cleared physically. That is to say the actual physical piece of paper had to be transferred to the issuing bank for it to be authorized for payment. This was done by flying around big boxes full of paper checks, and consequently, with all the aircraft grounded the checks could not be moved around, and bank transfers became almost impossible.
Once the planes started flying again, the system recovered, but the government was quite rightly rather freaked out by this situation. And so they passed a new law called "Checking in the 21st Century" act, or sometimes just called Check 21. What this law did is authorized the use of scanned images of checks (called IRDs or image replacement documents) as a method of clearing, so that the image of the check could be sent electronically and the physical check did not have to be sent. The idea is that when the check was scanned it was immediately destroyed (a process called "truncation"). The law and associated regulations is a lot of technobabble describing the format and contents of these IRDs and the process for scanning, truncating, sending and clearing these checks.
So, in theory you could have the same check in more than one place because an IRD is legally the same as the actual physical check. FWIW, it happens all the time. If I deposit a check in my bank account by my banking app taking a photo of it, I am advised to keep it for sixty days, at which point there are actually two copies of that same check floating around.
FWIW, this is why you can now use your cell phone to deposit a check, or why you can deposit checks in ATM machines. All because terrorists murdered thousands of people on 9/11.