When you row-reduce a matrix, the dimension of the column space stays fixed, so if $A,B$ have the same reduced echolon form then the dimensions of the column spaces are equal, but the column spaces might not be equal:
$$A=\begin{pmatrix}1&2\\1&2\end{pmatrix}\hspace{10pt}B=\begin{pmatrix}1&2\\2&4\end{pmatrix}$$
The have the same reduced echolon form, but different column-spaces.
In general, the only way to make sure that two matrices have the same column space is to column-reduce them (unless both are of full rank).
A matrix is not just as an array of numbers. It is helpful to think of it as a device for takes a vector as input and produces another vector as output by multiplication: that is for input $v$, the output is $Av$. This output is obtained by taking linear combination of column vectors of $A$, the coefficients for the linear combination being provided by the components of the vector $v$. So the output belongs to the column space.
It is possible that $Av$ is the zero vector,in that case $v$ is said to be in the nullspace.
For left multiplication $vA$, again one has similar interpretation, but everything in terms of rows of $A$ instead of the columns.
Now look at a matrix like $A=\pmatrix{1 & 2 & 3\cr 1 & 2 & 3 \cr 1 & 2 & 3\cr}$. Any scalar multiple every column vector is of the form $(x, x, x)^T$, and so linear combination would also be of the same type. So column space consists exclusively of vectors of the kind $(x,x,x)^T$. But any vector in the row space of $A$ is clearly of the form $(y, 2y, 3y)^T$. So column space and row space have nothing in common except the zero vector.
When the rank is 3, the columns form a basis for $\mathbf{R}^3$, and so every vector
is in the column space, including those in the row space, and vice versa.
When the matrix is symmetric then also we cna check row space and column spaces coincide. Ohterwise they don't. Only thing we can say is those spaces have the *same dimensions$ which is much different from saying they are the same.
Best Answer
This fails even in one dimension: $1$ and $2$ have the same column and null spaces. You can easily find other examples in higher dimensions. For example $I$ and $2I$.
In fact, all invertible matrices have the same column and null spaces, yet there are many different invertible matrices.