We can say that yes, it is a programming language.
According to man bash, Bash is a “sh-compatible command language”. Then, we can say a “command language” is “a programming language through which a user communicates with the operating system or an application”.
From man bash:
DESCRIPTION
Bash is an sh-compatible command language interpreter that executes commands read from the standard input or from a file. Bash also incorporates useful features from the Korn and C shells (ksh and csh).
http://www.gnu.org/software/bash/
Bash is the GNU Project’s shell. Bash is the Bourne Again SHell. Bash is an sh-compatible shell that incorporates useful features from the Korn shell (ksh) and C shell (csh). It is intended to conform to the IEEE POSIX P1003.2/ISO 9945.2 Shell and Tools standard. It offers functional improvements over sh for both programming and interactive use. In addition, most sh scripts can be run by Bash without modification.
And a UNIX shell is… http://en.wikipedia.org/wiki/Unix_shell
A Unix shell is a command-line interpreter or shell that provides a traditional user interface for the Unix operating system and for Unix-like systems. Users direct the operation of the computer by entering commands as text for a command line interpreter to execute, or by creating text scripts of one or more such commands. Users typically interact with a Unix shell using a terminal emulator, however, direct operation via serial hardware connections, or networking session, are common for server systems.