The answer is no. In order to upgrade your NikaTime account and continue using the NikaTime service after the free trial, you do not have to have a paid account Slack or Microsoft 365 account.
You may use NikaTime 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 NikaTime.
You may also use NikaTime on a free Microsoft Teams account without any limitations.