David Hansen wrote

record is destroyed when the book is handed back and the ticket and
card separated. However, in a computerised library this method of
operation is not used, instead the record of what books were
borrowed by who is kept by the system permanently. It may be that
the local staff do not have access to screens which will show this
record, but it is still there.

Kept for six months for the system that a lot of UK libraries use. The
information is available to the reader online which can be handy. I
imagine the business case was to detect vandalism and removal of
illustrations for sale especially now there is self-service terminals
for borrowing and return.

Personal data so Data Protection Act applies - if it's kept longer than
a reasonable time that's a violation.

--
Mike D