Comprehensive stats state that more than 50% of men have had extramarital affairs. Those affairs are carried out in secret and with deception in mind.
We're all human, after all, and lust and passion affect all of us to varying degrees.
Acquiring a "secret" phone in a false ID, that no-one knows about, is one way of keeping an affair reasonably secret.
The phone was registered to a woman. Doesn't that sort of defeat the purpose of using a fake name, if you're trying to hide an affair from the wife? Seems that if he was trying to hide a girlfriend, HE would have bought the burner phone in his name (or in a male name) and given it to her.