$'…' is a ksh93 feature that is also present in zsh, bash, mksh, FreeBSD sh and in some builds of BusyBox sh (BusyBox ash built with ENABLE_ASH_BASH_COMPAT). It isn't present in the POSIX sh language yet. Common Bourne-like shells that don't have it include dash (which is /bin/sh by default on Ubuntu among others), ksh88, the Bourne shell, NetBSD sh, yash, derivatives of pdksh other than mksh and some builds of BusyBox.
A portable way to get backslash-letter and backslash-octal parsed as control characters is to use printf. It's present on all POSIX-compliant systems.
esc=$(printf '\033') # assuming an ASCII (as opposed to EBCDIC) systemcol_green="${esc}[32;01m"Note that \e is not portable. It's supported by many implementations of printf but not by the one in dash¹. Use the octal code instead.
¹ It is supported in Debian and derivatives that ship at least 0.5.8-2.4, e.g. since Debian stretch and Ubuntu 17.04.