I think it is, because the specified path is relative to the current page (login is at an other level of path-nesting* than admin/manage).
A trivial but bad solution would be adding ../ for the css of admin/manage. But this soultion has the drawback, that you always need to adjust the ../ when you change the level of path-nesting* for an page.
To overcome this problem, make the path in the c:url-tag start with an /! (this make the tag aware that the url is context relative (relative to http://local.host:8080/test/), and the tag will automatically addhttp://local.host:8080/test` in front.
<c:url value="/css/style.css"/>
will be rendered to : http://local.host:8080/test/css/style.css
For the link use this way
<c:url value="/css/style.css" var="cssUrl"/>
<link rel="stylesheet" type="text/css" href="${cssUrl}"/>
*I do not know the correct term for "path nesting" at the moment - fell free to change it