I work at Stripe. This isn't really competitive with Square—very different customers.
It's clear that a card present product, which I believe is new for Stripe, is in direct competition with Square. Even if right now it's not being marketed towards the same customer base as Square, it's clear that with a few tweaks of the app plus a change in the manufacturing process, you could easily create Square-like POS systems. Why would Stripe leave out such a huge, proven market when it's clear this product is a stepping stone towards full POS systems?
Square: Small/individual merchants, Etsy style physical shops, trade shows, etc.
Stripe: large retailers, grocery stores, stores with many locations etc.