What is this ... electrical bike's setup? Is there a particular brand or model name for this? how to replace its battery?
It looks to be an older Currie Electro Drive, they were originally made in 1996 i believe. I don't know much to anything else about them though other than that is one and they exist.

Here is the companies website which offers support options. I would look for a serieal number and call them with questions about battery replacement.

http://www.currietech.com/about.html

Also, i found this random page that sells parts, you may be able to source a battery and spare parts from them as well.

https://electricscooterparts.com/currieelectrodriveelectricbicyclekitparts.html

This looks to be quite similar as far as the motor mount and rack go.

From what i can tell they make a very large battery pack that slides down the side of that rack and connects to those two metal terminals above the bicycles axle (the blue block). You can see them if you follow the second link i listed.