Did phonological systems (namely bopomofo) standardize Chinese phonemes?
As far as I know, Bopomofo and other schemes were based on the pronunciation of Beijing dialect. That is, Bopomofo and pinyin are a representation of the phonological inventory of Mandarin as pronounced in Beijing. I don't believe that this involved standardising 'two or more characters that had slightly different phonemes or sounds and/or tones' with a single bopomofo (or later Pinyin) to 'make things simpler'.

Of course, the creation of Bopomofo and pinyin involved drawing distinctions that were not found in other dialects of Chinese (including other pronunciations of Mandarin). One example is the distinction that pinyin draws between 'sh' and 's', which is missing from many dialects, even northern dialects (modern-day Dongbei, for example). It is therefore also possible that the opposite case was true, i.e., some distinctions drawn in other dialects were not drawn in the pronunciation of Beijing, and therefore ignored in Bopomofo and pinyin. I'm not knowledgeable about non-Beijing phonological inventories and have no way of commenting on this, but it is conceivable that the choice of Beijing dialect as standard (rather than, say, Nanjing or other varieties) may have involved getting rid of certain phonological distinctions found in other varieties of Mandarin. This was not caused, however, by the adoption of Bopomofo, but by the choice of a particular variety as standard.

Pronunciation of individual words has also been manipulated quite extensively in pinyin usage in order to bring about standardisation (I'm not sure about Bopomofo). For example, 法国 'France' was originally pronounced fàguó; it has now been standardised as fǎguó. 往 was originally pronounced wàng; it is now standardised as wǎng. There are many other cases where pronunciations have been standardised, including the abolition of alternative readings. But this is quite a different matter from making changes to the phonological inventory of the language.

So I think the answer to your question must be that, no, neither Bopomofo nor pinyin involved simplifying Mandarin phonological distinctions (where Mandarin is defined as based on Beijing pronunciation) out of existence.