If they have the ID software then yes they can, but you can get a blocking app that just shows your name as Private or with some other mobiles here in the UK, it will come up as Witheld.Either way your name and personal details will remain private.
If they have caller id than yes. Almost all phones have caller Id these days. What you can do is dial *67 then dial the number you want to call. It will show as Private caller on the other persons phones.