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 ...'.