This paper establishes the asymptotic error distribution of the tamed Euler method for stochastic differential equations (SDEs) with a coupled monotonicity condition, that is, the limit distribution of the corresponding normalized error process. Specifically, for SDEs driven by multiplicative noise, we first propose a tamed Euler method parameterized by $α\in (0, 1]$ and establish that its strong convergence rate is $α\wedge\frac{1}{2}$. Notably, $α$ can take arbitrary positive values by adjusting the regularization coefficient without altering the strong convergence rate. We then derive the asymptotic error distribution for this tamed Euler method. Further, we infer from the limit equation that among the tamed Euler method of strong order $\frac{1}{2}$, the one with $α= \frac{1}{2}$ yields the largest mean-square error after a long time, while those of $α>\frac{1}{2}$ share a unified asymptotic error distribution. In addition, our analysis is also extended to SDEs with additive noise and similar conclusions are obtained. Additional treatments are required to accommodate super-linearly growing coefficients, a feature that distinguishes our analysis on the asymptotic error distribution from established results.