Do animals' facial expressions have similar meanings to humans'?
I would say no, facial expressions in humans do not generally relate to facial expressions in other animals. Even though there are some cases (as described in the comments an previous answers) that refer to domesticated animals, I do not know of any convincing evidence that the facial expressions of humans are faithfully interpreted, or even mimicked, by other animals - at least outside the group of great apes. Further, the cases arguing that (domesticated) animals can do it are often anecdotal (I am not saying wrong!). Taking an evolutionary perspective, I would not even expect it to be happening (or rather by very close relatives only, i.e. other great apes) because facial expressions as behavioural signals are part of communication and therefore often rather species-specific.

But not only that ... there is emerging evidence that facial expressions are not even interpreted uniformly in all human populations: in this article by Crivelli et al. (2016) in Front. Psychol. the authors argue to include anthropological data to cognitive studies. According to their findings, not including a human evolutionary perspective might result in strong biases. They strengthen their point by giving an example of differential interpretation of facial expressions in a Melanesian society which is described in more biological terms in PNAS [Crivelli et al. (2016)]. What they found is that adolescents of this society interpret a facial expression (gasping face) that in other societies conveys fear as a threatening and angry expression.

Generalising this means that if there is divergence in signaling by facial expressions even among human populations, I would consider it unlikely that there is a general homologous system of facial expressions among mammals. There might, however, be a trend in domesticated animals to understand human expressions and adapt behaviour according to them. It still is an extraordinary claim to infer from this that they can actually mimic these expressions, and I have not yet seen convincing evidence for that.