A 2-dimensional plane within a 3-dimensional space is a 2-dimensional manifold.
However, if we see the 2-dimensional plane simply as being embedded in a 2 dimensional space, and assume that there is no 3rd dimension at all, is this plane then still classified as a manifold?
And what if the 2-dimensional plane is unbounded in all directions, thereby being identical to the 2-dimensional space in which it is embedded, is it then still called a manifold?
(the definitions that I find suggest yes, but I'm not 100% sure).
This is a funny instance where the plain English connotation of your question and the mathematical interpretation can differ.
It is in fact true (Whitney's embedding theorem) that every manifold (for suitable definitions of manifold, some people count the long-line as a manifold) can be embedded in some higher dimensional Euclidean space.
So if we understand the question in your title to mean: is it true that every manifold is necessarily a submanifold of a higher dimensional space? The answer is yes.
Furthermore, in some introductory textbooks of differential geometry of curves and surfaces, a manifold is in fact defined to be a submanifold of a higher dimensional space. This is generally a conscious choice made to make the subject more tangible for beginning students, and in view of the first point, is not really any loss in generality.
However, it is true that most modern geometers and topologists consider manifolds devoid of an ambient embedding; that is at least in part because of the discovery (going back to Gauss and Riemann) that there are intrinsic quantities/qualities of manifolds that can be studied independently of any particular chosen embedding.
So if we interpret your question as asking whether "embedded as a submanifold of some higher dimensional space" is included as one of the defining conditions of a manifold, the answer is "frequently not".