Can different AWS accounts manage different subdomains?

How to I tell the master account to push requests for .testing.example.com down to the child account.

The requests are referred, not pushed, but you can achieve the desired outcome by delegating the subdomain to a different set of Route 53 servers from those that host the parent zone.

Look at the new hosted zone you created for testing.example.com. This can be in the same AWS account, a different AWS account... any AWS account. There's nothing here that is "account" related. This uses standard DNS configuration. The whole of DNS is a hierarchy. The global root can tell you where to find com, and the com servers can tell you where to find example.com, and it's nothing materially different for example.com to tell you where to find testing.example.com instead of giving you a direct answer.

Note the 4 name servers that Route 53 assigned to the testing.example.com hosted zone. Verify that they are all different than the ones assigned to the example.com hosted zone. (For any of them to be the same should be impossible, but verify this.)

Now, back in the example.com zone, create a new resource record, with hostname testing, using record type NS, and enter the 4 name servers that Route 53 assigned to testing.example.com, in the box below.

Now, when a request for testing.example.com and anything below it arrives at one of the Route 53 servers handling example.com, the reply will not be the answer from testing.example.com -- the reply will provide the requester with the 4 NS records associated with testing.example.com and an answer equivalent to "I don't know, but try asking one of these guys."

That's how it's done.