What are the full requirements to get a best buddy in Pokemon Go?

The second menu on the buddy screen shows the number of activities done with your buddy, not the number of hearts. It counts every interaction you’ve done with your buddy, including the ones that don’t result in a heart.¹

So while you have done 621 activities with your buddy, the number of hearts hasn’t reached the 300 heart requirement. Unfortunately (at the time of writing) there is no way to see the total number of hearts your buddy has acquired.

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¹ To easily test and confirm this, go to your buddy, check the number of snapshots, and then spam taking snapshots of your buddy. While you only received 1 (or 2) hearts, the number increased by the number of snapshots taken.