Membership is about belonging
You're asking about something deep and wide—connection, belonging, and community. If you want to understand the effect of "Member since …" statement, you have a bit of reading to do. Or you can skip to the last paragraph of this answer.
Belonging affects people in seemingly illogical ways, as Susan Weinschenk writes in her blog. If you want something more academic than a blog to read, start with this. Membership is one of the four elements of sense of community, according to David McMillan and David Chavis. Sense of community is tied to self-identity, according to Seymour Sarason. If you're looking at an eCommerce site, there's an additional connection between self-identity and consumerism that you can explore.
The statement "Member since …" is a way to reinforce connection (where the user might not have felt it as much) or belonging, and is a subtle way to say "Welcome back" that can unobstrusively be repeated frequently or seldomly.