Why does sed outputs "char 53: unterminated `s' command"

The s command in sed, uses a specific syntax:

s/AAAA/BBBB/options

where s is the substitution command, AAAA is the regex you want to replace, BBBB is with what you want it to be replaced with and options is any of the substitution command's options, such as global (g) or ignore case (i).

In your specific case, you were missing the final slash /, if you add it, sed will work just fine:

➜  ~  sed 's/database_name: [^ ]*/database_name: kartable_$ME/'
database_name: something
database_name: kartable_$ME

info sed 'The "s" Command' includes the full description and usage of the s command.

Missing / at the end.

sed -i "s/database_name: [^ ]*/database_name: kartable_$ME/" $PARAM_FILE