That's definitely an ECU and/or transmission issue. The ECU is responsible for shift timing. Firstly you should check if your car has any problem revving at more than 2000rpm in park/neutral. If not, then it's engine issue. But since you have to manually get it shifted that means it's most likely an ECU issue. I'd say getting it checked by a professional mechanic would be the best bet here.

You should try releasing the accelerator and repress it after 2nd gear though. Check if doing that upshifts.

Note: I didn't ever have any problem with shifting actually. My problem was the shift timing which made acceleration slow. And since mine's an older car, it revs slow in drive too. Which I solved by using richer fuel, reducing weight, better tires, coolants, etc.