I would like to build USB to Serial board that can program the known ATmega328P microcontroller based on ATtiny2313 due to its characteristics .
Is that impossible?
Not impossible by any means, but aside from the academic exercise, simply not worth the trouble and cost.
Whether you go for the "clones" or the "genuine" article for comparison, it will be cheaper to get a fully-fledged, ready built module using either the FTDI chip or the CP2102 (or that other one) which are designed specifically for the purpose and will perform reliably and comprehensively.
The Atmel chip on the UNO is likewise designed for the purpose. It is not the cheapest option, but used for the flexibility it permits. If you need such flexibility, you would use that chip, but if you merely need the USB to TTL function, better to use a chip - and ready-made board - that simply does the job (and cheaply).