Why is the Pinyin for 波 bo1 not buo1? And why is 多 duo1 instead of do1? Recently I started to learn Pinyin. The system is relatively easy to follow but some ambiguity does exist. For example, I found the usage of finals o and uo confusing: why the Pinyin for 波 is bo1, not buo1? and 多 is duo1 instead of do1? Would appreciate if someone could give the reason for such usage.
For all practical purposes, you can think of this as a spelling convention. The finals are pronounced the same, even if all finals are to some extent influenced by the preceding initial.
In Pinyin, there is (almost) no overlap between these two spellings, so any given initial that can be followed by -uo can never be followed by -o and vice versa.
The actual pronunciation is arguably also identical, even though this always sparks emotional arguments when mentioned to native speakers with no training in phonetics (they typically claim that there's an obvious difference). This could be either because they are influenced by orthography (the way the word is spelt influences how they perceive it) or that the coarticulation going on with the preceding syllable in combination with the spelling makes them take note of the difference more than they otherwise would have. The phenomenon that spelling influences how native speakers pronounce words is not uncommon, see for example this list on Wikipedia for examples when this happens in English.
In many narrow transcriptions of Mandarin syllables (such as the one in Duanmu San's The Phonology of Chinese (2007) or Lin Yen-Hwei's The Sounds of Chinese (2007), the -uo and -o finals are transcribed exactly the same way: [ᵂoo], e.g. [pᵂoo] for bo and [tᵂoo] for duo.
This means that the sound is treated as a glide, which in one case gets its own letter in Pinyin, but in the other cases (after b, p, m and f) is merely implied.
I don't know the historical reason for this, though. If you listen to the syllables spelt with only -o, there's a clear glide in there, which is also easy to see if you look at spectrograms in e.g. Praat.
Finally, I said at the beginning that there is almost no overlap between these finals. That's because we do have both wo (which could be thought of as uo) and o, which are clearly different sounds. Since none of these have a normal initial, maybe you could say there's overlap, even though they are spelt differently.
However, while the first is very common (e.g. 我), the second is very rare and could be argued to not be a standard syllable at all. It's normally only used for modal particles like 喔.
I have done several projects that involve recording native speaker audio for all possible syllables in Mandarin, and most recorders (educated native speakers) don't know what to do with o or simply read it as wo, further hinting that this is not a normal syllable.
For a clear distinction, check the Pinyin chart over at Chinese Pronunciation Wiki.
Possibly, lo also follows this pattern, which would create an overlap with luo, but then again, lo is used in a similar way to o and not really as a full syllable. It should be abundantly clear that bo, po, mo, fo are not pronounced as lo or o, further indicating that it's mostly about spelling.
I have answered several similar questions here and have collected other potential tricky Pinyin issues here: A guide to Pinyin traps and pitfalls: Learn Mandarin pronunciation.
References:

Duanmu, S. (2007). The phonology of standard Chinese. Oxford University Press.
Lin, Y. H. (2007). The Sounds of Chinese. Cambridge University Press.