This claim needs some real world evidence to back it up (and usually it's not about a performance impact, but instead a perceived image quality impact).
IME a lot of care is taken for compressing asset data, and if there would be a chance to reduce the asset size by another few percent without losing too much detail (that's the important part), it would be done. In the end, textures need to end up in memory as one of the GPU compatible hardware compressed texture formats (e.g. BCx) - which is important not only for reducing memory usage, but mainly for increasing texture sampling performance on the GPU (reduced memory bandwidth and better cache locality when reading texture data from GPU memory).
Those hardware-compressed texture formats rule out any of the popular image formats like JPEG, PNG etc... that might have a better overall compression, but are hard to decode on the fly to one of the hardware texture formats (but there are now alternatives like https://github.com/BinomialLLC/basis_universal), but even a generic lossless compressor (even good old zip) on top of BCx is already a good start.