WhatsApp is rolling out a new update that will block the ability to take a screenshot of your contacts' profile picture. Blocking screenshots of profile pictures on WhatsApp is apparently intended to prevent impersonation and other privacy issues.
Other messaging services that operate on multiple platforms, such as Telegram and Signal, do not currently offer this functionality. To further improve privacy, you can hide your WhatsApp profile picture by going to Settings > Privacy > Profile Picture, from there you can determine who will see your picture.