Judaism makes a clear distinction between the value of human life and animal life (see here and here); for this reason the comparison between pet and child that is part of the original question is not a helpful analogy.
There is no requirement that Jews who are observant of traditional Jewish practice must, or should, give up an existing pet or even avoid adopting a new pet. The fact that some newly observant Jews do give up their pets may be an expression of their association with a particular community of other observant Jews. That is, there are communities of observant Jews for whom having a pet is seen as inappropriate – this approach is not based in a prohibition by Jewish law, rather, it is a phenomenon of socialization and communal identification (identifiable groups include those of Middle Eastern culture and Haredim – though exceptions do exist). Still, there could be some halakhic/Jewish legal origin for such a phenomenon.
Traditional Jewish observance includes not touching any object on shabbat that has no utilitarian purpose on shabbat – these objects are called “muktzeh.” For example, since one is forbidden to write on shabbat the rabbis decreed that touching a pencil/pen is prohibited on shabbat – in this way, the sages wanted to “build a fence” around the Torah that would guard against people making mistakes either through forgetfulness or error. Pets as we know them today – dogs, cats, gerbils, birds… – are a relatively modern development. For earlier generations of humans in general and Jews in particular, the only animals of contact were farm animals such as cows, goats, chickens and horses. Since a requirement of sabbath observance is to allow one's animals to rest – no plowing, carriage riding, milking – the animals had no utilitarian purpose on shabbat and, thus, animals were categorized as muktzeh by our tradition. Accordingly, avoidance of pet ownership makes sense for those for whom animals have no utilitarian purpose on shabbat.
However, if a person finds a companion animal to be a source of joy, companionship, affection, play, support, or other benefit there is no absolute prohibition to such a relationship whether it be shabbat or any other days of the week. Companion animals are a relatively recent development in the Jewish world so it is understandable that questions arise regarding appropriate approach to the modern human-animal relationship. Whether one desires to absent companion animals from his personal realm or desires to include companion animals in his personal realm – both positions are available for the religious-minded practitioner. Rather than abandoning pets, it may be wise for pet owners to find like-minded communities that allow for the reconciliation of one's observance of Jewish practice to be integrated with one's approach of mutually beneficial relationship with companion animals.
Answered by: Rabbi Adam Frank