The answer is no. In order to upgrade your Nikabot account and continue using the Nikabot service after the free trial, you do not have to have a paid account Slack or Microsoft 365 account.
You may use Nikabot with a free Slack account - in that case, Slack will only display the most recent 10,000 messages and archive anything older than that, but that has no impact on using Nikabot.
You may also use Nikabot on a free Microsoft Teams account without any limitations.