Interesting. To me, the man pronouncing “o peixe” sounds different to the woman pronouncing it, differing not only in the initial consonant of “peixe”.
I feel the man says /ɔ 'pe:ʃɪ/, while the woman says /ɔ 'beɪʃi/. Maybe a difference in regional accent?
Please excuse my mixing up of brackets and slashes when using IPA. I should probably have used brackets here and in many of my previous posts.
I hear a “p” sound in both cases. But of course our ears haven’t been formatted in the same way.
Ermigerd. Those two sounded the same to me 
In that 1st video pronouncing Portuguese “peixe”, I can’t quite decide if I hear a /b/ or a /p/. It seems in between to me.
But in the 2nd video pronouncing (British) English “pain”, I can very clearly hear /pʰ/. I would never confuse that with /b/.
When you (as a native English speaker) pronounce “Spain” versus “pain”, don’t you notice a difference?
From my understanding, the first would usually be pronounced as /speɪn/ and the second as /pʰeɪn/.
I don’t know if there would be a difference between standard British English and standard American English.
I heard different sounds for Paixe as well. The second one sounds like what I expect from a p in other romantic languages, but I kind of heard a Chinese b in the first one. Almost a korean ㅃ.
Funny. Although I don’t speak Chinese at all, in the video with 2 different people pronouncing Portuguese “paixe”, I somehow thought the second one (/'beɪʃi/) sounded more Chinese than the first one (/'pe:ʃɪ/). Perhaps this is because the second one sounded quite similar to how I hear Chinese people pronounce the first syllable of “Beijing”.
But in your video about Chinese pronunciation of “b”, she indeed says it is unvoiced, i.e. /p/.[¹].
If that claim were true, then I wonder why that Chinese sound is not just transliterated as “p”.
But when she proceeds to pronounce “bàba”, I hear /pàba/, so it would make sense to transliterate it to “b” at least in medial positions.
This potential pattern of “b” becoming unvoiced /p/ in initial positions[²] seems very similar to what we are discussing here about Korean phonology. I wonder if there is some sort of sprachbund between Chinese and Korean. Although even to my ears, the initial consonant as pronounced here(“Beijing”) does sound like a /b/ (voiced), not like a /p/ (unvoiced), so my hypothesis about the pronunciation of initial “b” as /p/ in Chinese may not be valid.
[¹] It’s clear that this video is aimed at English speakers, because it recommends to pronounce “b” as the “p” in “speak”, which is unaspirated and unvoiced. Was the potential confusion of /p/ with /pʰ/ for English speakers part of the reason why the common Latin transliteration of the name of the Chinese capital (/beɪˈd͡ʒɪŋ/) changed from “Peking” to “Beijing”?
[²] By the way, in Dutch “b” becomes unvoiced (/p/) in final positions. We pronounce “lab” as /lɑp/ and “bob” as /bɔp/.
Some romanizations transliaterate it as a “p”.
So WG transliteration of Mandarin Chinese used p for /p/ and pʼ for /pʰ/, which was officially replaced by pinyin in mainland China in 1958, that uses b for /p/ and p for /pʰ/.
These developments seem very similar to the MR transliteration of Korean, that used p for ㅂ(/p/) and p’ for ㅍ(/pʰ/), which was replaced by RR in South Korea in 2000, that uses b for ㅂ and p for ㅍ.
Both MR and RR have some additional rules about the transliteration of ㅂ in specific situations.