sid wrote:

I am trying to finish my VFR, but my student certificate expires after
24 months.
The examiner informs me that the next student certificate needs to be
issued by my instructor.

One common confusion is that your student certificate is in fact two
documents on one piece of paper: the student certificate proper and the
medical certificate, each with their own expiration date; it so happens
that the AME can issue both, but the student certificate part can also be
issued by the FSDO or an examiner (see 14 CFR 61.85); as far as I know,
your instructor cannot renew it (unless s/he is also an examiner)

--Sylvain