Yes, Bitly Links are case sensitive. This means if your link is "bit.ly/CustomWord," then "bit.ly/customword" will lead to an error page or to a different destination.
You can make both work, however, if you edit your link to the other options. This is because when you edit a link, the older versions of that link still work. For example, to make both of the links above point to the same destination, first shorten a link to bit.ly/CustomWord, then edit it to bit.ly/customword.
Editing your link combines metrics for all versions of the link.
What about the URL for a Bitly Page?
The URL for your Bitly Page is also case-sensitive. This is the link you create to share your Link-in-bio page with your audience.
You can change the URL for your Bitly Page if you don't like its capitalization. Just be aware that this doesn't work like a redirect. You'll need to update all links to your Bitly Page wherever you've shared it.