Yes, they had tank names, but on a more ad-hoc basis than e.g. the official Squadron names used by the British. They appear to be the exception rather than the rule.
Tactical markings are usually the division symbol on the right hand side front and rear panels, facing towards the tank, and possibly the panzer regiment tactical symbol on the left, although these became rarer as the war progressed.
Otherwise it is turret numbers, most common is the three number system (company/platoon/vehicle) but there were other variations.
Some units had various odd turret heraldry, but these were usually formation specific.
National markings were balkan crosses, generally on the hull sides for Panthers.
Late war vehicles were often pretty plain (turret numbers and crosses only) as vehicle attrition rates were extremely high by 1944.
This model of a Panther G is fairly typical: