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Trello is Software as a Service - you don't download Trello itself, you use a client to use Trello from the Cloud. It is built for browsers, but you say "Trello App", which implies you're looking at the main dedicated client for it.
That is a free download and does not need a login or account to download and install.
But it does need to be told which Atlassian account you want to connect to your Trello with. I guess it's asking you for those details when you first run it, so it can log into your Trello. As you've already got an Atlassian account, you should probably use that.
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