When reviewing a manuscript for the second time, is it fair to comment on aspects I did not realise the first time?
I believe that it is entirely reasonable to make major requests of a manuscript on second review, as long as it is maintaining your basic standard for acceptance rather than moving the goalposts.

For example, I have had the experience where things about a paper were very unclear to me in the initial review because of shortcomings in the authors' presentation.  Upon revision, those things became much clearer---but not in a good way.  The extra information shed light on serious flaws in the authors' work, which has led me to in some cases recommend further major revision and in other cases recommend rejection.

On the other hand, some reviewers seem to like asking authors to do entirely new work, not because it is necessary for publication, but because the reviewer wants to know the answer or thinks it will make "a more interesting paper."  This happens most frequently with "glamour" journals.  I believe this sort of "moving the goalposts" on publication is not appropriate even in the first review, and doubly so in a revision review.