If $m,n \to \infty$ with $\frac{m}n \to \lambda \in (0,\infty)$ (let's suppose that $n$ depends on $m \in\mathbb{N}$), and if you are saying that each $\mathbf{B} = \mathbf{B}_m$ is an $m\times n$ random matrix whose entries are i.i.d. drawn from a fixed (across $n$) probability distribution $\mu$ with zero mean and unit variance, then the behavior of $$ \frac1m \mathbf{B}\mathbf{B}^\dagger $$ is known to be potentially very far from $\mathbf{I}_m$, and even much worse its inverse. This is the content of the Marcenko-Pastur law, https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution.

If $m,n \to \infty$ with $\frac{m}n \to \lambda \in (0,\infty)$ (let's suppose that $n$ depends on $m \in\mathbb{N}$), and if you are saying that each $\mathbf{B} = \mathbf{B}_m$ is an $m\times n$ random matrix whose entries are i.i.d. drawn from a fixed (across $n$) probability distribution $\mu$ with zero mean and unit variance, then the behavior of $$ \frac1m \mathbf{B}\mathbf{B}^\dagger $$ is known to be potentially very far from $\mathbf{I}_m$, and even much worse its inverse. This is the content of the Marcenko-Pastur law, https://en.wikipedia.org/wiki/Marchenko%E2%80%93Pastur_distribution

On the other hand, products of matrices of the type you described appear in multivariate Fisher statistics, which is covered in Chapter 2.5 in a great book by Yao et. al., Large sample covariance matrices and high-dimensional data analysis. A revised edition can be found here: https://www.researchgate.net/publication/272093238_Large_Sample_Covariance_Matrices_and_High-Dimensional_Data_Analysis