Local Bitcoin is just a website were any user can register as an buyer and seller. They usually do not monitor each profile and also do not interfere in the trade requirements of the buyer or seller.

Due to the increasing number of crypto scam attempts any buyer or seller could ask for an identity verification to be safe at his side if he is scammed later.

If you were working on an PayPal realted trade on local Bitcoin you are most probably need to do a Identity verification with the trader as its an reversable method of payment and is risky. I personally have dealt with users asking for identity verification for trades mostly PayPal related.