Single Engine Pilot reported a possible airspace violation and NMAC . The pilot cited the complicated airspace at this location and confusion with ATC instructions as contributing to the event.

Narrative

On a southbound departure from ZZZ I requested clearance from ZZZ2 Tower to transition west through the ZZZ2 Class D airspace. The controller approved my request with a midfield crossover restriction. She called [small jet] traffic 2 miles south of ZZZ3 at 3;000 feet. During her call she responded with an incorrect call sign. Subsequent to my initial response I visually acquired the [small jet]. By this point I had initiated a west bound turn and climb to higher altitude for my transition as I believed I was well within the ZZZ2 Class D and below the 3;000 foot floor of the Class B.The controller became excited and issued instructions to stop climb to 'XYY' and subsequent transmissions to 'YY'. This caused extreme confusion to me as to whom her instructions were directed. The airspace in the location near my westbound turn is complicated as there is overlapping design with different control and altitude jurisdictions. I use aircraft GPS navigation and EFB data for a visual presentation of airspace. While it is possible I penetrated Class B airspace it was purely unintentional. At one point the Tower controller stated 'you almost hit a [small jet']. I had visual contact with the small jet who was going to pass well behind my flight path. The complicated nature of the airspace design combined with the controller's tone and delivery of incorrect call sign contributed to an extremely confusing experience.