Ubuntu includes GNU tar, which recognizes the format by itself! One command works with any supported compression method, per the manual.

tar xf archive.tar.xz
tar xf archive.tar.gz
tar xf archive.tar

etc. If tar gives a Cannot exec error, you may need to sudo apt install xz-utils first.

Since you are not allowed to use outside libraries, you are not restricted to a specific format of the tar file either. In fact, they don't even need it to be all in the same file.

You can write your own tar-like utility in C# that walks a directory tree, and produces two files: a "header" file that consists of a serialized dictionary mapping System.IO.Path instances to an offset/length pairs, and a big file containing the content of individual files concatenated into one giant blob. This is not a trivial task, but it's not overly complicated either.

If file filename.tar.gz gives this message: POSIX tar archive, the archive is a tar, not a GZip archive.

Unpack a tar without the z, it is for gzipped (compressed), only:

mv filename.tar.gz filename.tar # optional
tar xvf filename.tar

Or try a generic Unpacker like unp (https://packages.qa.debian.org/u/unp.html), a script for unpacking a wide variety of archive formats.

determine the file type:

$ file ~/Downloads/filename.tbz2
/User/Name/Downloads/filename.tbz2: bzip2 compressed data, block size = 400k