In a lecture on connections it was claimed that a manifold "attains a shape" when it is equipped with a connection while there were no mentions of a metric.
Is it correct to intuitively think that the connection determines the "shape" via how tangent spaces are "connected" or equivalently a rule how tangent vectors are moved in parallel manner between such spaces, while the metric only determines length of the same tangent vectors(and hence any curve).
And therefore the metric structure and connection structure is completely independent in general, i.e two completely separate things, unless your connection is defined in terms of the metric by kind of brute force as in the Levi-Cevita case, which is a very special case yet very meaningful.
Here's a link to the lecture https://www.youtube.com/watch?v=2eVWUdcI2ho. The discussion starts at around 5:30 and lasts until about 9:00.
This statement is only meant to provide intuition, since the notion of "shape" is undefined. A (Riemannian) metric structure and a connection are different things indeed, which is quite obvious once you look at their definitions. Few things to remember:
An (affine) connection on a manifold need not preserve any Riemannian metric.
Two different connections can correspond to the same Riemannian metrics.
Two different Riemannian metrics can have the same Levi-Civita connection.
An affine connection does determine a curvature tensor $$ (X,Y,Z)\mapsto R(X,Y)Z, $$ but in order to convert such tensors into functions (on the manifold or on some bundles over the manifold) you need an extra piece of data, which is usually taken to be a Riemannian (or semi-Riemannian) metric.
There are connections on much more general bundles over manifolds (for instance, a bundle over a surface whose fibers are again surfaces); such connections, at the first glance, do not at all look like connections which are discussed in the lecture.
Edit. On the second thought, one way to interpret the notion of "shape" (say, for surfaces) is that "shape refers to the sign of curvature" (at a point or on some open subset). If your surface is (locally) isometrically embedded in $E^3$ then the sign of the curvature determines if the surface (locally) is "ellipsoidal-like" or "saddle-like", which one may think of as the shape of the surface. One can prove that the sign of the Gaussian curvature (say, at a point) is determined by the Levi-Civita connection. More precisely, if two metrics have the same connection, then they have the same sign of curvature. At the same time, a connection cannot determine the magnitude of the curvature. One can see this by multiplying the metric by a positive constant: Connection does not change, but the curvature does (unless it is zero, of course).
One can prove a similar statement about sectional curvature (in higher dimensions). In many ways, sign of the curvature is more important than the magnitude.