No, gnome-terminal is not meant to behave differently than the others, although the exact shade of the 16 base colors is somewhat different across all the terminal emulators. (Update based on JdeBP's comment below: in some palettes, e.g. Solarized, these colors are deliberately totally different.)
You should look around in gnome-terminal's Edit -> Profile Preferences -> Colors dialog and choose a scheme that matches your taste.
In the escape sequence \e[01;32m 01 stands for bright and/or bold (there's a confusion about it across terminal emulators) and 32 stands for green foreground. 38 on the other hand means the terminal emulator's default foreground, which is green if you happen to have the "green on black" color scheme by default, but more typically it's either white or black.
Assuming your shell is bash, its PS1 prompt is required to enclose all the escape sequences between \[ and \], this is how bash knows that printing those won't advance the cursor, and hence editing the command line won't fall apart. Assuming that these are indeed present in pairs in PS1, bash consumes them and never prints them to the terminal, so there's no way gnome-terminal could do anything nasty there. Probably they are not properly balanced in your PS1.