1. I want to raise a cat.

2. I want to keep a cat.
3. I want to have a cat.
4. I want to grow a cat.
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Can we use all the verbs to have the same meaning? Does 'grow' mean 'raise' here?

We don't usually talk about "growing a cat."

The others are fine.

We grow tomatoes and sweet corn and apples. We don't grow animals.