No; it depends on the individual package whether it is open source of nor. Ubuntu uses a lot of Open Source software (including the Linux Kernal) and packages but is not entirely open source. It derives it’s own package base from Debian, and then adds it’s own flavour to things as well as commerical tools it pushes.

Linux Mint is an Ubuntu derivative; its sounds like the Indian Government would be doing the same thing. Basically like Mint, they would use Ubuntu and it’s packages as the basis of their system and rely on most of it to be updated & maintained by Ubuntiu’s teams upstream, but then build their own repositories that contain other software or their own perferred modified versions of things originally taken from Ubuntu. They can build a version of Linux that they control including what software is installed, and when it is updated.

They would not have to make their distribution freely available, but if they modified any open source packages they would have to make those available as open source packages (depending on the license of the open source software). However that can be very difficult to inforce, and if it’s done in a closed military system you’d never even know a modified version of the software exists if they chose not to share it.

Although Ubuntu contains a lot of “open source” software, it doesn’t mean the Indian Governments version would necessairly be open source. But the big benefit to India would be potential complete control of any part of the software chain, and no reliance on big tech companies like Microsoft for the OS and core software like Office. That saves a lot of money and is also potentially more secure (in a national security sense), depending on how much people trust the US government not to interfere in american Tech companies. There has been talk of forcing backdoors into US software in the past which would make any big nation nervous about being reliant on their software.