Why do I have access to Springer articles/books using a laptop when I am not logged in?
It looks like Springer does this using cookies.
I also observed what you described in the question. I was able to access articles that had a paywall when I was neither in my university's IP range nor logged in. However, at the footer of the page, near the copyright notice, it showed name of my university and a mention that this was what gave me access to that particular page.
This did not work while browsing incognito, which suggested the use of cookies. I was then able to verify this by clearing cookies from link.springer.com in my browser, after which I lost access.
I suspect the cookie was stored at some point when you were in the institutional IP range, and had a sufficiently long validity that you continue to have access, and will probably do so for as long as it is valid or until you remove it.
Edit: I was able to verify this as well, accessing Springer from my institution's VPN resulted in me getting access again, which persisted even after leaving the network. A further check showed that the cookie is a session cookie, valid till the end of the session. The duration of storage of such cookies varies by browser and operating system, but should generally persist for at least as long as you keep the browser open (which on a laptop could potentially be a long time), and often even beyond system restarts (e.g. Firefox, Chrome).