How to extract paired reads from two paired-end reads file?

Firstly, extract sequence IDs of two file and compute the intersection:

$ gzip -d -c read_1.fq.gz read_2.fq.gz | seqkit seq --name --only-id | sort | uniq -d > id.txt

Then retrieve reads using id.txt:

$ gzip -d -c read_1.fq.gz | seqkit grep --pattern-file id.txt  | gzip -c > read_1.f.fq.gz
$ gzip -d -c read_2.fq.gz | seqkit grep --pattern-file id.txt  | gzip -c > read_2.f.fq.gz

Note that this example assumes that the IDs in the two reads file have same order. If not you can sort them after previous steps. Shell sort can sort large file using disk, so temporary directory is set as current directory by option -T ..

$ gzip -d -c read_1.f.fq.gz | seqkit fx2tab | sort -k1,1 -T . | seqkit tab2fx | gzip -c > read_1.f.sorted.fq.gz
$ gzip -d -c read_2.f.fq.gz | seqkit fx2tab | sort -k1,1 -T . | seqkit tab2fx | gzip -c > read_2.f.sorted.fq.gz

The question is more : what did you do you that created orphan reads ? Did you trim/quality filter your r1 and r2 reads independently ? Some trimming tools, like trimmomatic, have a paired-end mode to avoid those issues.

But if you really need to fix those files, you can try to remove the orphant reads.