Can't help you on the Tesco front, but that does sound very much like an auditor doing the classic "I'm here to audit against this standard and only this standard".
If you're genuinely a separate site/business (irrespective of proximity) then it seems a bit OTT to extend an audit from your neighbour into your facility.
My response to these NCs would be that the root cause was the client not advising you that they have a standard that applies here, and that your corrective action is therefore to request a copy of the applicable standard(s) to review for potential implementation.
This makes the point that the if they want a specific standard then they need to tell you what it is, and give you chance to fully review it, before they audit against it. If you go through the request process and they still send you the manufacturing standard then again you can challenge them and explain that you're not a manufacturing facility - it may be that they can then agree a sensible position on which parts of the standard can be excluded.
I've been in a not dissimilar position with clients springing requirements on us during an audit. It's usually a large company as they are the most compartmentalised - the auditor will often be completely separate to the department with whom you interact on a daily basis - and thus there is potential for them to be completely unaware that you have not been sent any details about what is expected.
Last year we received our first major NC in years in this type of circumstance. The auditor insisted they couldn't do anything as they were tasked solely to audit to their standard, so we had to accept it on the day, and then challenge it afterwards. They did accept that they were in effect the root cause, and rescinded the NC completely, but it required some odd conversations!