How to add timestamp(month_year) to the file after matching specific string?

$ is special to the shell within double quotes as it's used for parameter expansion (like in $var, $1, $channel_name), arithmetic expansion (like $((1+1))) and command substitution (like $(cmd) which you are actually using here).

$ is also special in regular expressions (like on the left-hand side of the s/regex/replacement/ sed command) as it's the operator that means "end of the subject", unless you use -E/-r to enable extended regular expressions, it's not special unless the $ is at the end of the regexp itself or followed by \) (or \| where supported). Still, it's a good habit to escape it.

Both the shell inside double quotes, and sed in its regexps support \ as an escaping operator to remove the special meaning of characters. \ can also escape itself in both cases.

So, here, you could do:

sed -i "s/\\\$channel_name Policy/\$channel_name Policy_$(date +%m_%Y)/g" filename

That is, \\\$ to escape \ itself and \$ for the shell so that \$ is passed to sed where \ escapes $ again. On the right hand side \$ is enough as $ is not special in the replacement part of the s/regexp/replacement/ command.

Though not POSIX, this would also work in any shell that I know:

sed -i "s/[$]channel_name Policy/&_$(date +%m_%Y)/g" filename

[$] is another way to escape $ in a regexp. [...] matches on a set of characters, here comprising only one: $. While for the shell [$] is unspecified by POSIX, I don't know of any shell that doesn't leave it as is. You'd need [\$] for that shell code to be POSIX compliant though.

We're also using & in the replacement which stands for the string that was matched by the regexp.

Another option would be to use single quotes for most of the string (inside which $ is not special), and double quotes for the command substitution, still escaping the $ in the regexp even though it's not strictly necessary:

sed -i 's/\$channel_name Policy/&_'"$(date +%m_%Y)/g" filename

Note that you need $(...) to be double quoted as otherwise split+glob would be applied on it.

As commented by 'don_crissti', you have to escape the dollar sign otherwise it tries to get a variable content (which is empty in your case).

Alternatively it would be safer if you use a different separator than '/' as you also have slashes in your file. However it is not an issue in your case as long you do not use them in the replacement expression.

My working solution:

sed -i 's!\$channel_name Policy!&_'$(date +%m_%Y)'!g' policies.txt

The second possible improvement is the usage of the & in the replacement as it is exactly the searched expression plus some extra.

And third, in my test I was only able to make the command $(date +%m_%Y) working when outside of the sed expression (wrapped by single quotes).

I tried in GNU sed (the above line) and with OSX sed as well but none of them where able to interpret properly the $(command).