Is concept of "bit" in computer programming similar to the concept of "bit" in information theory?
They are not the same, but they're related. In particular, if you look at a computer memory holding $M$ "computer" bits, where each bit can be considered random and independent of all other bits, and there are roughly 50% of zeros, then the memory also holds roughly $M$ "information theory" bits.

Of course, this is often not the case: computer bits are usually correlated, and not uniformly random. This is why they can be compressed. Compressor programs such as LZW ("source coders" in information theory parlance) work, in a sense, by making each computer bit hold one information bit.

Edited to add: This example may make the distinction clearer. Consider a memoryless source with two outputs, $m_1=000$ and $m_2=001$, with probability 0.5 for each. Clearly, the information in each message is one (information) bit, but its length is three (computer) bits. A source coder, such as the Huffman algorithm, will readily code the messages to $c_1=0$ and $c_2=1$, compressing the source output. You can easily extrapolate this example to a source that produces ASCII-encoded text.

Note that, in the case of written languages in general and English in particular, nobody knows what the actual source entropy is, because there is no model for it. That is why there are contests for the best compression of large bodies of text; nobody is really sure what the optimum compression algorithm for English is.