Say $a=b$. Is "Do the same thing to both sides of an equation, and it still holds" an axiom?

This axiom is known as the substitution property of equality. It states that if $f$ is a function, and $x = y$, then $f(x) = f(y)$. See, for example, Wikipedia.

For example, if your equation is $4x = 2$, then you can apply the function $f(x) = x/2$ to both sides, and the axiom tells you that $f(4x) = f(2)$, or in other words, that $2x = 1$. You could then apply the axiom again (with the same function, even) to conclude that $x = 1/2$.

"Do the same to both sides" is rather vague. What we can say is that if $f:A \rightarrow B$ is a bijection between sets $A$ and $B$ then, by definition

$\forall \space x,y \in A \space x=y \iff f(x)=f(y)$

The operation of adding $c$ (and its inverse subtracting $c$) is a bijection in groups, rings and fields, so we can conclude that

$x=y \iff x+c=y+c$

However, multiplication by $c$ is only a bijection for certain values of $c$ ($c \ne 0$ in fields, $\gcd(c,n)=1$ in $\mathbb{Z}_n$ etc.), so although we can conclude

$x=y \Rightarrow xc = yc$

it is not safe to assume the converse i.e. in general

$xc=yc \nRightarrow x = y$

and we have to take care about which values of $c$ we can "cancel" from both sides of the equation.

Some polynomial functions are bijections in $\mathbb{R}$ e.g.

$x=y \iff x^3=y^3$

but others are not e.g.

$x^2=y^2 \nRightarrow x = y$

unless we restrict the domain of $f(x)=x^2$ to, for example, non-negative reals. Similarly

$\sin(x) = \sin(y) \nRightarrow x = y$

unless we restrict the domain of $\sin(x)$.

So in general we can only "cancel" a function from both sides of an equation if we are sure it is a bijection, or if we have restricted its domain or range to create a bijection.

There is a way in which it is an axiom, and another in which it isn't. I'll try to describe both.

When you do formal logic and start with variables ($x_1,...,x_n,...$), relation symbols ($R_i, i\in I$) and function symbols ($f_j, j\in J$). You build out your formulas with these. An important notion is that of a term in this language. A term is defined recursively as either a variable, or a string of the form $f_j(t_1,...,t_n)$ where $f_j$ is a function symbol of arity $n$ and $t_1,...,t_n$ are terms (this is purely syntactical).

Then you build a deduction system that consists in rules that you may apply in certain situations (for instance if you proved $A$ and $B$, you can prove $A\land B$).

One of these rules is the substitution rule: one way to define it is the following : for any terms $t_1,...,t_n, u_1,...,u_n$ and any function symbol $f$ of arity $n$, if for all $i$, $t_i = u_i$ is proved then one may deduce $f(t_1,...,t_n) = f(u_1,...,u_n)$

In this situation it is an axiom.

However in the common situation of algebra and "the working mathematician", it is a consequence of another substitution rule, the substitution rule for relation symbols. Indeed, all maths can be built from set theory with no function symbol and only one relation symbol ($\in$). In this setting a function $A\to B$ is defined (for instance) as a subset $f$ of $A\times B$ such that for all $x\in A$, there is a unique $b\in B$ such that $(a,b)\in f$. $f(a)$ is then defined as this unique $b$

Now if $x=y \in A$, $f:A\to B$ is a function, then $(x,f(x)) \in f$ and $(y,f(y))\in f$ and so by the substitution rule for relation symbols $(x,f(y))\in f$, so that $f(x)=f(y)$ (by uniqueness).

The substitution rule for relation symbols is very similar to the one for function symbols and is, again an axiom: it can be described as: if $t_1,...,t_n,u_1...,u_n$ are terms, $R$ is a relation symbol of arity $n$; if for all $i$, $t_i= u_i$ has been proved and $R(t_1,...,t_n)$ has been proved, then one may deduce $R(u_1,...,u_n)$

(you may see that the rule is not so far from an axiom, but technically it's not one in this second situation)