Can you animate a logo in Final Cut Pro?

As Ben said, Motion is the most adequate tool for that. But you can try doing it in FCP, it is just not as easy.

You can add one or more Draw Masks and keyframe them to make the logo appear to draw itself.

If you don't have Motion, maybe you can make do with it.

OR... Maybe you can do it, much more easily, using... Keynote.

Here is an example, done in Keynote in less time than it takes to write this reply. This is just ONE of many effects that are built-in in Keynote, maybe there is one that serves you well.

It depends on just what you want to do with it. You can easily move it around, resize it, distort it, make it appear or disappear, for example.

Perhaps you tell us more about what you want to achieve, perhaps even link to an example video showing something similar, so we can better advise. It’s possible, depending on what you want, that other tools may be easier or more adequate. Please elaborate.

As Luis says, it is possible to create an animation similar to the linked example using only FCP. I think you can use effects such as

・Masks

・Key frames

・Contrast, etc.

I'm creating an animation, but it's a lot of troublesome work.😅

You'd want Motion for that.