Extending your definition of ${^{b}}a$ to have it be just $a$ for $b = 1$, so ${^{b}}a$ is $a^{({^{b-1}}a)}$ for $b \ge 2$, then your equivalence relation becomes

$$a^{({^{b-1}}a)} \equiv a^{({^{b}}a)} \pmod{10^{\operatorname{len}({^{b}a})}} \;\to\; a^{({^{b-1}}a)}\left(a^{({^{b}}a - {^{b-1}}a)} - 1\right) \equiv 0 \pmod{10^{\operatorname{len}({^{b}a})}} \tag{1}\label{eq1A}$$

In particular, consider

$$a^{({^{b}}a - {^{b-1}}a)} - 1 \equiv 0 \pmod{2^{\operatorname{len}({^{b}a})}}, \;\;  a^{({^{b}}a - {^{b-1}}a)} - 1 \equiv 0 \pmod{5^{\operatorname{len}({^{b}a})}}\tag{2}\label{eq2A}$$

As you mention in your [comment](https://mathoverflow.net/questions/471657/prove-ba-equiv-b1-a-pmod-10-lfloor-log-10-ba-rfloor#comment1224875_471657), it's relatively trivial to show that $10 \nmid a$, so have that be an added restriction. In addition, since $a = 5$ is a solution, also have $a \neq 5$. Next, since $\operatorname{len}(a^a) \ge a - 1$ for all $a \ge 2$, then my [answer](https://math.stackexchange.com/a/4863149/602049) in the Math SE you linked to proves $a = 5$ is the only solution for $b = 2$. As also shown in that other answer, both equations in \eqref{eq2A} don't hold for any of those other values of $a$, but at least one equation must hold for \eqref{eq1A} to be true.

Assume neither equation in \eqref{eq2A} is true for any $a$ under consideration for $b = k$ for some $k \ge 2$, so \eqref{eq1A} is also not true. With $b = k + 1$, consider one or both equations in \eqref{eq2A} depending on if $2$ or $5$ is a factor of $a$. Using the [Lifting-the-exponent lemma](https://en.wikipedia.org/wiki/Lifting-the-exponent_lemma) (LTE lemma), with the first one if $2 \nmid a$, since $p = 2$ and the exponent is even, we get

$$\nu_2(a^{({^{k+1}}a - {^{k}}a)} - 1) = \nu_2(a - 1) + \nu_2(a + 1) + \nu_2({^{k+1}}a - {^{k}}a) - 1 \tag{3}\label{eq3A}$$

Using that $\nu_2(a - 1)$ or $\nu_2(a + 1)$ is $1$, let $c$ be the value of $a - 1$ or $a + 1$ where it's $\nu_2$ value is $\gt 1$. Also, from the induction hypothesis in \eqref{eq2A}, we have $\nu_2({^{k+1}}a - {^{k}}a) \lt \operatorname{len}({^{k}a})$. Thus, the RHS of \eqref{eq3A} then becomes

$$\nu_2(c) + \nu_2({^{k+1}}a - {^{k}}a) \lt \nu_2(c) + \operatorname{len}({^{k}a}) \lt \operatorname{len}({^{k + 1}a}) \tag{4}\label{eq4A}$$

If $5 \nmid a$, then with the second equation in \eqref{eq2A}, we need to consider $a \mod{5}$. If that modular value is $1$, we can just use $a$, else with $2$ or $3$, we need to use $a^4$, while if it's $4$, then $a^2$ is required. Let $j$ be the minimum required exponent. If $j \nmid {^{k+1}}a - {^{k}}a$, then the second equation in \eqref{eq2A} doesn't hold. Otherwise, let ${^{k+1}}a - {^{k}}a = jm$, so we then have

$$\nu_5\left((a^{j})^m - 1\right) = \nu_5(a^j - 1) + \nu_5(m) \tag{5}\label{eq5A}$$

Similar to \eqref{eq4A}, we get

$$\nu_5(a^j - 1) + \nu_5(m) \lt \nu_5(a^j - 1) + \operatorname{len}({^{k}a}) \lt \operatorname{len}({^{k + 1}a}) \tag{6}\label{eq6A}$$

This shows that both equations in \eqref{eq2A} don't hold for $b = k + 1$ and, thus, \eqref{eq1A} also doesn't hold. As such, this proves by induction that the only solution to your equivalence relation is as you stated, i.e., $a = 5$.