You can do this with $where, by creating RegExpfor string1, and then checking it with string2:
db.test.find({$where: 'RegExp(this.string1).test(this.string2)'})
, MongoDB 3.4+, , $indexOfCP:
db.test.aggregate([
{$project: {foundIndex: {$indexOfCP: ['$string2', '$string1']}, doc: '$$ROOT'}},
{$match: {foundIndex: {$ne: -1}}},
{$replaceRoot: {newRoot: '$doc'}}
])
, $redact:
db.test.aggregate([
{$redact: {
$cond: {
if: { $eq: [{$indexOfCP: ['$string2', '$string1']}, -1]},
then: '$$PRUNE',
else: '$$KEEP'
}
}}
])