Does the mse loss equal the L2 loss in an autoencoder?

I have been trying to implement an variational autoencoder on a texture dataset. For that I used the VAE example given by Keras as a base and changed the layers to my liking. Since the example uses binary crossentropy loss I wanted to change it to the L2 loss because a Paper regarding a texture autoencoder suggested so.

I have searched how to implement the L2 loss, but did not quite find a clear answer. Is the mse loss equal to the L2 loss in the context of the variational autoencoder?

Yes, $$L_2$$ loss is another name for squared error. Not in autoencoders, but in general. $$L_2$$ in the name comes from the norm in mathematics. In different places you would see either sum of squared errors, or their mean, but they are technically equivalent, while mean is preferred since it has units independent of sample size.