Is it necessary to use a specific switch controller for a flyback converter circuit?
You have answered your own question, as the DA2077 transformer is directly compatible with the NCP1027 PWM controller.
The DA2077 is NOT directly compatible with the simplistic TNY290 PWM controller.
The reason is the boost winding the transformer has to self-power the PWM IC once it starts up. This isolated winding also isolates the high voltage from the output voltage.
Both the DA2077 and the TNY290 detect the output voltage using an opto-coupler for isolation, connected to the 5 volt output.
To some degree these PWM IC's are flexible with their designs, but not pre-made transformers where the winding ratios are fixed.
To put it another way, the transformer could be used with the TNY290, but its output voltage and efficiency cannot be determined to be safe or as expected. There would be one winding not used. The transformer may not work properly at the 132 KHZ frequency the TNY290 puts out. This frequency is fixed internally to the IC.
Here is a quote from the DA2077 document:

The DA2077-AL flyback transformer has been specifically developed for
use with ON Semiconductor’s NCP1027 Standby Power Supply for ATX
Applications and is specified in Application Note AND8241/D.

This is a quote from the TNY290 documentation:

Oscillator
The typical oscillator frequency is internally set to an
average of 132 kHz. Two signals are generated from the oscillator: the
maximum duty cycle signal (DCMAX) and the clock signal that indicates
the beginning of each cycle.