Vector spaces are also groups, with addition as the group operation.
Subgroups correspond to subspaces.
In the context of vector spaces, your question is:
Let $H$ and $K$ be subspaces of a vector space $G$ which have a common element besides the origin. Does this mean that either $H$ or $K$ is a subspace of the other?
The problem is now more geometrical than algebraic, and perhaps your intuition will work better.
The answer is no: Let $G=\mathbb R^3$ and $H$ and $K$ two different two dimensional subspaces.
Since your original claim fails for the special case of vector spaces, it is false.
Linear algebra is sometimes a good way to check abstract algebraic ideas.
Notice, however, that vector spaces are always abelian as groups, and not even all abelian groups are vector spaces.