This is an example of formal written English. The word 'should' is not replacing 'if' . You can tell that it is not replacing 'if', because if the sentence contained the word 'if' it would be '..if the user CHOOSES to..' - not 'choose'. In fact, the sentence is using an alternative and rather formal inversion construction in which the word 'if' is not necessary. A more standard construction would be this: "The browsers provide a mechanism to store a password if the user should choose to have the password remembered by the browser." The 'should choose' form is a kind of subjunctive, suggesting that this is just a possibility. When we use this construction, you can invert the modal verb (should) and the subject (the user) to make a conditional form that doesn't use 'should'. This construction is quite common in polite business letters. For example, you might come across an ending like this: 'Should you have any further queries, please do not hesitate to contact us.'. This a neater way of saying 'If you should have any ...'.

In formal English, when "should" and "were to" are used, "If" can be omitted. But yes, you can also say it like "If the user chooses to save their password, the browser provides a mechanism to store it". It doesn't sound quite as nice though. I think that the former version would be much more common in technical writing.