Before I can answer a brief digression is necessary.
Start of digression.
Some particular "interpretations" of quantum mechanics make the division you describe, but the division is neither necessary nor desirable. Different interpretations of quantum mechanics tend to fall into two categories. Some simply deny the theory's implications (e.g. - Copenhagen), others say that it is false and call for its replacement (e.g. - GRW). The former ought to be rejected out of hand. If the implications of quantum theory disagree with reality then we ought to face that squarely and reject the theory, if it is true then it is important that we should understand it to be able to use it to predict and understand stuff. Theories that disagree with quantum mechanics ought not to be called interpretations of it for clarity.
When QM is applied to macroscopic objects, quantum theory implies that multiple versions of those objects exist that can't interfere with one another as a result of decoherence. Those versions are sorted into layers that are approximately autonomous from one another and approximately obey the laws of classical physics. These layers are called universes and the whole of physical reality is called the multiverse. The multiverse has structure in addition to the universes, so it is not accurate to describe the multiverse as just a collection of parallel universes:
End of digression.
You ask whether a measuring instrument can just be another particle. If two systems interact with one another and not with anything else they will be entangled and some distinctively quantum mechanical effects can occur. You can make them interact in such a way that each individual particle will be able to undergo quantum interference. And they can exhibit Bell-type correlations of a sort that are much celebrated, but usually very badly misunderstood:
Interference involves a system starting out with one of its observables sharp: that observable has the same value in all of the versions of the particle involved in the experiment. The observable becomes unsharp, undergoes some motion and becomes sharp again in a way that depends on the motion it underwent while it was unsharp. If you try to make a record of that observable's value while the particle is unsharp, then you prevent the interference. For more details see
A measurement should be described as an interaction that makes a record that will be preserved indefinitely since that makes it easy to use the measurement result for experimental tests. Since you can undo the interaction between the two particles and erase the record, that doesn't seem to fit the bill. And the existence of permanent records, not some alleged classical level in physics explains why you don't see quantum mechanical effects with objects like coffee mugs.
So the reason to not regard an interaction with another particle as a measurement is a mixture of physics and methodological limitations on what should count as a record.