grep -E "([[:digit:]])[[:space:]]?\1"

this command is working on my local VM but not passing the test cases. What is the mistake here?

what a nice question!

Not sure why this works in test case0 but fails when i submut, for hidden test case: grep -E '([0-9])\1{1,}'

why doesn't this work?

grep -E '(.) ?\1'`

nor does this: grep '\(.\) \?\1'

Since the grep in this environment doesn't support Perl-style regular expressions, you can't use "\d". If you use Basic Regular Expressions (the default and equivalent to --basic-regexp or -G), you can use the character class [0-9] (see https://pubs.opengroup.org/onlinepubs/9699919799/basedefs/V1_chap09.html).

In BRE, the capture group must be defined in a subexpression with \( \) and then can be recalled with \1 (see https://en.wikibooks.org/wiki/Regular_Expressions/POSIX_Basic_Regular_Expressions). Finally, BRE doesn't support using the precise '?' matching operator to pick out the optional space, but it does permit the more greedy '*'. So you end up with:

grep '\([0-9]\)\s*\1'