There are several approaches I'd take to this.
Firstly you ask how to measure what is there. I think this could be relatively simple and I wouldn't wash it off. When you wash metal filings, they tend to sink to the bottom of whatever they are rinsed onto and resist further rinsing away. I would take a damp cloth and wipe the magnet, then, if necessary, photograph it.
I don't think I can answer on "what is acceptable" because my question which follows is, for milk, why are you getting any? It's an interesting thing that finding metal has become normalised it seems in your process. Why is that?
Personally I think this is a bit back to front. You should be defining in your HACCP plan what is and isn't acceptable in terms of foreign matter risk including size. So that should be giving you your release decision, however, what you are detecting is an early warning in your process that something is wrong. It is not normal to find metal pieces on magnets routinely unless for example, you have a root vegetable product where it's conceivable that some fragments would come in with soil, however, even then, you'd question washing processes not just accept it. This needs to be investigated and resolved.
For investigation, I'd use a recognised RCA technique like 8 step problem solving or at least fishbone.
I'm not sure if you have an equivalent organisation to Campden BRI or RSSL in Vietnam but those are the kinds of places in the UK which could do metal analysis and support an investigation into where it's come from. But there is obviously also stuff you can do on site.
- Inspect machinery for signs of scratches and wear and do this full end to end. Concentrate of course on any areas which are moving.
- Check the finds with a normal magnet, not rare earth to see if it's mild steel or iron (unlikely to be part of your process).
- Check filters are intact and in place and an appropriate gauge for your process.
- Look for any recent maintenance work and see what the process is in general for cleaning up after cutting or grinding.
- Look at cleaning processes. I had an incident once with metal where we found the root cause was how we were washing some removable items, they were bashing each other in a sink so we bought a frame to clean them in.
I'm sure you can think of far more potential root causes. But I'm old enough to have seen enough incidents where a level of failure becomes the norm and accepted. Then you later find there was an issue in your process and you missed the early warning.