Keep me logged in (for up to 30 days) says the en.wikipedia.org login page, but it never happens for me anymore, regardless of browser. It does not happen on Internet Explorer, Mozilla Firefox, Google Chrome, Opera, or Safari. Making sure cookies are turned on does not seem to help. In 2013 it worked fine. I am using Windows 7.
Topic on Project:Support desk
You should raise your problem there: Wikipedia:Village pump (technical)#Unable to remain logged-in, specifying since when you have the problem, and see if more people have that particular problem. May be worth to report a bug for that, too.