It is annoying, and they're definitely pushing people toward invasive smartphone apps using various means, but this particular annoyance has a good reason:
Browsers simply aren't as secure as individual apps, mainly because they execute code from other web sites as well. That means credentials available to a browser are only one remote exploit away from being compromised. And browsers are big, complex beasts with an unending stream of vulnerabilities waiting to be discovered and exploited. Tight countermeasures make sense for things as important as banks and medical info.