What is the error covariance structure for linear mixed effects models estimated by lme4?

Yes, this is correct. You can, e.g., read this off from the conditional distribution of the response given the random effects shown in expression $(2)$ in this lme4 vignette. Such an error distribution is called "spherical".

Addendum

lme4 can fit linear mixed models obeying measurement models of the form $$ \boldsymbol{\mathcal{Y}} = \boldsymbol X \boldsymbol \beta + \boldsymbol Z \boldsymbol{\mathcal{B}} + \boldsymbol o + \boldsymbol \epsilon, $$ with response vector $\boldsymbol{\mathcal{Y}} \in \mathbb R^n$, fixed-effects design matrix $\boldsymbol X \in \mathbb R^{n \times p}$, fixed-effects parameter vector $\boldsymbol \beta \in \mathbb R^p$, random-effects design matrix $\boldsymbol Z \in \mathbb R^{n \times q}$, random-effects vector $\boldsymbol{\mathcal{B}} \in \mathbb R^q$, (known) offset vector $\boldsymbol o \in \mathbb R^n$, error vector $\boldsymbol \epsilon \in \mathbb R^n$;
and the distributional assumption $$ \begin{pmatrix} \boldsymbol{\mathcal{B}}\\ \boldsymbol \epsilon \end{pmatrix} \sim \mathop{\mathcal N_{q + n}} \left( \begin{pmatrix} \mathbf 0\\ \mathbf 0 \end{pmatrix}, \begin{pmatrix} \boldsymbol \Sigma_{\boldsymbol \theta} & \mathbf 0\\ \mathbf 0 & \sigma^2 \boldsymbol I_n \end{pmatrix} \right), $$ with positive semi-definite covariance matrix $\boldsymbol \Sigma_{\boldsymbol \theta} \in \mathbb R^{q \times q}$ parameterized by the covariance parameter vector $\boldsymbol \theta \in \mathbb R^m$.

Such linear mixed models can also be written in the hierarchical form $$ \boldsymbol{\mathcal{Y}} \,|\, \boldsymbol{\mathcal{B}} = \boldsymbol b \sim \mathop{\mathcal N_n} \left( \boldsymbol X \boldsymbol \beta + \boldsymbol Z \boldsymbol b + \boldsymbol o, \sigma^2 \boldsymbol I_n \right), \\ \boldsymbol{\mathcal{B}} \sim \mathop{\mathcal N_q} \left( \mathbf 0, \boldsymbol \Sigma_{\boldsymbol \theta} \right), $$ which is the form used in the lme4 vignettes.

From this we can see that the implied distribution of the error vector $\boldsymbol \epsilon$ is $\mathop{\mathcal N_n}\left(\mathbf 0, \sigma^2 \boldsymbol I_n\right)$, a spherical multivariate normal distribution.