Military doesn't need same quality to monitor, Starlink needs to also transmit and receive data(albeit slow speed), a military sat would just need to listen for a signal and know active and possibly identify some characteristics which combined other data sources Signal Int will have access to could be used to uniquely identify.

Starlink satellites are designed to be cheap(<$1M) and small in weight (~1,000Kg) and size(fit 20-60 in Falcon 9), and lot of the innovation is manufacturing at scale.

I am skeptical they come anywhere close to purpose built military satellites which costs upwards of $300-$400M[1] and are much larger in weight (5,000Kg-10,000+) and size (use full fairing ) and fly DH, Atlas as well and lever vertical integration. Given what we know about KH-11s, Trump photo tweet, the tech in the donated mirrors etc Military seem generation ahead in production tech not just experimental ones.

[1] There is some pork for sure in all government contracts, but not $299M worth.