I work at a company that is one of Plaid's largest customers and have chatted with their engineers before at a meetup in San Francisco. They were very open that they store the data themselves, and even shared this blog post with me about how they poll for data:
https://blog.plaid.com/distributed-duplicate-detective/.
For what it's worth, I actually like Plaid's API compared to Yodlee, but have serious concerns about privacy. There's no way to determine if they are _just_ collecting transactions when you request the user's transactions as a dev. The user is handing over their user/pass/MFA and Plaid could be doing god knows what with that.