I really like some websites when they use a mascot or an animal as part of the logo. But when is it really okay to use them.
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Animals used in a logo have the certainty for a start-up and the commitment for an existing organization. If intelligently used within logos, their impact is far greater than that of a colorful logo with typography.
There are definitely situations where the branding or the image of the company fits in quite well with the use of an animal or mascot as you have pointed out. All those examples are aligned to the company/business branding or image in some ways. For example, MailChimp resonates with a personal touch that is also friendly, while HostGator is linking the name with the mascot (mountaingoat software is another example).
Another particular reason for using a mascot may have to do with the history of the company, or a personal pet that is close and dear to the owner that is incorporated as part of the startup and eventually becomes the history. IPEVO is one company that uses animals in their marketing and advertising as well, and I think the husky from Trello might be another example (may need to verify this).