That common character is called a generation name, as it is shared among the same generation in a family. This practice is seen much less commonly in recent times. For example, none of my friends bear a name in this format.

However, sometimes while siblings share a common character in their name, it may not a generation name. Some parents like to name their children this way, though in such case, the common character may appears in any position of the given name. So it's kinda hard to tell whether it's a generation name thing just by the name, without consulting the name bearer themselves or their parents.

Interesting enough, my family uses the (less common) format of <FamilyName><RealName><WordFromList>.

My mom's family follows this convention.

Yes that is the format. Sometimes, there can be two character names. I only know that my grandparents are from shanghai, but I cannot speak shanghainese. I am learning from my dad and aunts because the intonation is more difficult for me as compared to speaking chinese or cantonese.

You might be interested in Tatoeba's collection of Shanghainese audio recordings: https://tatoeba.org/eng/audio/index/wuu