My Macbook Pro's Mail app continues to play the bass sound when a mail message is received, even though I've turned off all incoming mail sounds and notifications (see screenshots below). Is there anything else I can do to eliminate the incoming mail sound? I'm on a 2018 15" Macbook Pro running 10.14.5 and Mail 12.4 (3445.104.11).
I have had a similar issue. I use SpamSieve and I was unaware (or had forgotten) that there is a preference within SpamSieve to make a sound after a message is identified as 'good'. That is why the Basso sound (the default from SpamSieve) was being played after the 'New Message' sound - that, in experimenting, I had chosen in Mail.
Are you using SpamSieve? Or perhaps another filtering service that rides on Mail?