Multiplication by One

As Ross Millikan has pointed out, mathematicians like to call multiplicative identities $1$ because we like things to behave familiarly.

There are exceptions to this though, especially when we're working with sets of numbers that are endowed with structures other than the ones we usually use. The easiest example that comes to mind is the set $S=\{0,2,4,6,8\}$ with addition and multiplication defined modulo $10$ (in other words, if I multiply or add numbers and they exceed $10$, then I take the remainder dividing by $10$. So $6+8\equiv 4$ modulo $10$ since $14=10+4$, while $4\cdot 8\equiv 2$ modulo $10$ since $4\cdot 8=32=3\cdot 10+2.$) If you sit down and multiply each element of $S$ by $6$, you will find that $6$ is the multiplicative identity, not $1$ (which isn't even in the set).

This is kind of cheating though because in some sense $S$ with this structure is "the same" as $\{0,1,2,3,4\}$ with addition and multiplication defined modulo $5$, and in this case $1$ is is the multiplicative identity.

Outside of the usual integers / rational numbers / real numbers that we learn about in grade school, the symbol "$1$" is often used as a placeholder for the multiplicative identity in a given algebraic structure, such as a ring or field.

To build such a structure, we start with some set, which we'll call $S$. Next, we define "operations" on this set, which are functions that input any two elements of $S$ and output a single element in $S$. Typically, we call these operations "addition" and "multiplication" and use the standard $\times$ and $+$ symbols (though these can be defined quite differently from grade school addition/multiplication). Finally, the mathematician will specify a list of axioms that this set and the operations should satisfy, such as associativity, commutativity, and so forth.

Among these axioms will often be a statement along the lines of "There exists a special element in $S$, which we will call the multiplicative identity and denote by the symbol "$1$" which has the property that $y \times 1 = 1 \times y = y$ for any element $y \in S$.

I don't think there's really any context where we want to relax the axiom $1a = a$. But sometimes, we want to relax the axiom $0a=0$. I've seen the terminology almost-semiring used to describe algebraic structures where addition and multiplication behaves as expected, except that $0a = 0$ might not be true. Almost-semirings arise naturally: for example, suppose $X$ is a set, $S$ is a semiring, and consider the collection of all partial functions $X \rightarrow S$. This is an almost-semiring, but if $X$ is non-empty, it won't satisfy $0a=0$. To see this, let $a$ denote any non-total function $X \rightarrow S$. The big difference between $1a=a$ and $0a=0$ is that $a$ occurs exactly once on each side of $1a=a$ (it's a "balanced" identity), whereas this isn't true of $0a=0$. This means that the identity $1a=a$ can be expressed using operads, whereas $0a=0$ cannot. The end result is that while most ways of building constructions will preserve the identity $1a=a$, many will not preserve $0a=0$. For another, similar example, try doing algebra in the powerset of a semiring. You'll quickly notice that $1A=A$ holds, but that $0A=0$ doesn't.

It may not be what you're looking for, but I believe this is related. In the case of limits, repeated multiplication by a limiting value of $1$ can have surprising effects.

Consider this: $$\left(1+\frac{1}{n}\right)^{n}=\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)\left(1+\frac{1}{n}\right)\ldots$$

For large values of $n$, this appears to approach: $$\left(1+0\right)\left(1+0\right)\left(1+0\right)\ldots=1\cdot1\cdot1\ldots=1$$ $$1^{\infty}\stackrel{?}{=}1$$

However, this is not the true value of the limit. In reality: $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=e\gt1$$

It appears that infinite repeated multiplication by $1$ somehow becomes another value entirely. This is why $1^\infty$ is considered an indeterminate form.

Infinity is a special case. People underestimate the concept of infinity (even those who know higher math!). It is best defined as something unreachable (a limit), and it makes sense that the normal rules break down in weird ways when you try to play with it.

Of course there are people who disagree, but that is beyond the scope of this question