TSA is supposed to ensure the person presenting the boarding pass at the PSC is the person named on the BP - so asking for the child's name is reasonable. But as noted, there is no requirement to provide documentary proof of the child's identification/name. Children are much less likely to lie to an adult, let alone one in uniform with a shiny badge, than is an adult so directly asking the child to provide the name is part of the drill. Asking for age is either the TSO being conversational / trying to appear "friendly" or a deliberate effort at screening for abduction / child trafficking (where the responses might indicate referral for additional screening).

Most people will go with the flow and let the child answer. But some will stick to principles and push back by not allowing the child to answer more than just the name. And you are correct - taking that tack might well result in being referred to additional screening - it's TSO discretion. I have done it both ways and depending upon how tight my connections are or how important it is to get where I am going (or truth be told - pressure from my wife not to push back on them) I am with how you and FUTR suggest handling it.