replace capture match with capture group in bash GNU sed - sed

Question

I've looked around to find a solution to my problem in other posts listed bellow, but it looks my regex is quit different and need special care:
How to output only captured groups with sed
Replace one capture group with another with GNU sed (macOS) 4.4
sed replace line with capture groups
I'm trying to replace a regex match group in big JSON file,
My file has mongoDB exported objects, and I'm trying to replace the objectId with the string:
{"_id":{"$oid":"56cad2ce0481320c111d2313"},"recordId":{"$oid":"56cad2ce0481320c111d2313"}}
So the output in the original file should look like this:
{"_id":"56cad2ce0481320c111d2313","recordId":"56cad2ce0481320c111d2313"}
That's the command I run in the shell:
sed -i 's/(?:{"\$oid":)("\w+")}/\$1/g' data.json
I get no error, but the file remains the same.
What exactly am I doing wrong?

Answer 1

Finally I've managed to make it work, the way regex works in bash is different then in regexr.com tester tool.
echo '{"$oid":"56cad2ce0481320c111d2313"}' | sed 's/{"$oid":\("\w*"\)}/\1/g'
gives the correct output:
"56cad2ce0481320c111d2313"
I found it even better to read from stdin and output to file, instead of writing first to JSON file, then read, replace and write again.
Since I use mongoexport to export collection, replace the objectId and write the output to JSON file, my final solution looks like this:
mongoexport --host localhost --db myDB --collection my_collection | sed 's/{"$oid":\\("\\w*"\\)}/\\1/g' >> data.json