grep -P no longer works. How can I rewrite my searches?

If your scripts are for your use only, you can install grep from homebrew-core using brew:

brew install grep 

Then it's available as ggrep (GNU grep). it doesn't replaces the system grep (you need to put the installed grep before the system one on the PATH).

The version installed by brew includes the -P option, so you don't need to change your scripts.

If you need to use these commands with their normal names, you can add a "gnubin" directory to your PATH from your bashrc like:

PATH="/usr/local/opt/grep/libexec/gnubin:$PATH"

You can export this line on your ~/.bashrc or ~/.zshrc to keep it for new sessions.

Please see here for a discussion of the pro-s and cons of the old --with-default-names option and it's (recent) removal.

If you want to do the minimal amount of work, change

grep -P 'PATTERN' file.txt

to

perl -nle'print if m{PATTERN}' file.txt

and change

grep -o -P 'PATTERN' file.txt

to

perl -nle'print $& while m{PATTERN}g' file.txt

So you get:

var1=`perl -nle'print $& while m{(?<=<st:italic>).*(?=</italic>)}g' file.txt`
var2=`perl -nle'print $& while m{(property:)\K.*\d+(?=end)}g' file.txt`

In your specific case, you can achieve simpler code with extra work.

var1=`perl -nle'print for m{<st:italic>(.*)</italic>}g' file.txt`
var2=`perl -nle'print for /property:(.*\d+)end/g' file.txt`