I would claim that such structures are imaginary, not structures at all, and the whole field is a sect (a socially constructed movement based on sectarian consensus which regards "knowledge" of details of abstractions, which does not make any sense outside of sectarian contexts).

Let's talk the most basic algebraic laws. Yes, there is indeed no difference in adding 2 apples to 3 apples, or adding 3 of them to 2. The question is from which pile to start, and the result is the same. (Multiplication, being just repeated addition, is also obvious - there is really no difference which side of a rectangle comes first).

Notice, that there is still a fundamental connection to reality. Adding C to O has exactly the same properties. There is no difference which comes first.

The identity element is more tricky, because nothing of this sort existed in nature. Nature has distinct start and stop sequence, and does structural pattern matching in the literal sense.

So, we would superimpose an element upon reality which lacks it, the same way as it goes with zero. Let's say, that trying to add what's can't be added (by physical properties) produces no change to the original element, and it is equivalent to addition of a zero.

Generalising this noop we will get something similar to identify.

1 for multiplication, is natural , while an identify matrix is an artificial construction.

Okay, this is all common sense. The important thing that there is literally nothing deeper than this shaky generalisation which we call monoid. First, because only addition and multiplication are "true monoid".

Protein production by an enzyme is almost it, but no identity.

There is no identities outside your head. A list is a monoid because of added '() - the empty list, which, again, does not exist anywhere. There is no such thing as an empty DNA sequence, empty molecule, etc.

I could go on, but maybe you already see where it goes. Everything is imaginary and too abstract to have any applicable, meaningful context.