I think that's more like "Let's remember some guys". The app/website is like "Oh yeah I remember that device, good times, good times" and then forgets.
@gedeonm I hate to say it, but if you're mostly a Safari user, Safari may be the problem. I think it clears out stored cookies if you haven't visited a website in more than a week.
I know that for paying bills it renders any "Remember this device" stuff absolutely useless – I only visit those once a month. :-/
One specific shopping site I use will log me out after an hour (or less), no matter how many times I check "stay logged in". 😡