Yes, exactly. Maybe literally, because they still have their hunting instincts. They even bring their preys (mostly birds and mouses) to you to train and educate you. Tigers are also cats. They are just/only a little bit bigger.

There is an interesting article about cats from 2016. It says the following:

According to a study, around four billion birds are killed by cats yearly in the USA. Also considering the other small animals they hunt, we get a shocking number: almost 30 billion!

The article: Why Do Cats Bring Their Prey To You?