Warning : This answer is wrong, it assumes a too restricted (almost trivial) concept of equivalent codes. See Jyrki Lahtonen's comment below. I'll delete it as soon as it's un-accepted.
To add to GitGud's comments: the most relevant common feature of two equivalent codes is that they have the same codebook (set of codewords, which is the row space of $G$).
Hence, any property of some linear code $c_1$ with generator matrix $G_1$ will be shared by another equivalent code $c_2$ with generator matrix $G_2$, as long as that property is inherent to the codebook alone (for example: minimum distance).
A property that is not preserved by equivalent codes is the mapping of raw inputs to codewords - of course, elsewhere the two codes would be not just equivalent but identical.