Not necessarily. The abbreviation "ESOP" is ambiguous. There are at least 8 variations I know of:
- Employee [ Share / Stock ] [ Owership / Option ] [ Plan / Program ].
You'll find references on Google to each of those, some more than others. For fun you can even substitute the word "Executive" for "Employee" and I'm sure you'll find more. Really.
So you may be mistaken about the "O" referring to "options" and thereby implying it must be about options. Or, you may be right. If you participate in such a plan (or program) then check the documentation and then you'll know what it stands for, and how it works.
That being said: companies can have either kind of incentive plan: one that issues stock, or one that issues options, with the intent to eventually issue stock in exchange for the option exercise price.
When options are issued, they usually do have an expiration date by which you need to exercise if you want to buy the shares.
There may be other conditions attached. For instance, whether the plan is about stocks or options, often there is a vesting schedule that determines when you become eligible to buy or exercise.
When you buy the shares, they may be registered directly in your name (you might get a fancy certificate), or they may be deposited in an account in your name. If the company is small and private, the former may be the case, and if public, the latter may be the case. Details vary. Check the plan's documentation and/or with its administrators.