Errors appear when the Shannon sampling series is applied to approximate a signal in real life. This is because a signal may not be bandlimited, the sampling series may have to be truncated, and the sampled values may not be exact and may have to be quantized. In this paper, we truncate the multidimensional Shannon sampling series via localized sampling and obtain the uniform bounds of aliasing and truncation errors for functions from anisotropic Besov class without any decay assumption. The bounds are optimal up to a logarithmic factor. Moreover, we derive the corresponding results for the case that the sampled values are given by a linear functional and its integer translations. Finally we give some applications.