Does this extension require planning permission ?

That looks to be to the rear, behind the rear building line.

But agree, if to the side, planning required unless it’s a garage/shed/store etc

Retention Permission required should they wish to regularise it.

How did you "find" the plan?

Is it your house? Do you want to regularise it or has somebody said something to you?

In a professional Surveying context?

You should be asking the vendor to provide a copy of the Final Grant of Planning Permission along with copies of the Certificates of compliance with said planning permission and building regulations compliance. May also require BCAR confirmations.

Or alternatively, a certificate/opinion on compliance with the exempted development regulations (which technically, they should not be able to provide).

Spanner in the works. What if the hatched area is the original house and the extension is all to the rear?

Can you show us what the original house looked like?

More questions than answers.
The OP states there are no windows on the walls that face the boundaries!

Ohh I know, hence the confusion from info drip feeding.

A declaration of exemption is not satisfactory for conveyance though. It’s not actually called a declaration. It’s an answer to a question based on documents lodged as part of a section 5 application. The OP will still need to issue an opinion of compliance with the exempted development regulations using the section 5 application decision as grounds for the cert.