This sounds a bit like the schwa we find in many other
languages, like English, German, Dutch, Hebrew, French.
But this Portuguese sound is very different, darker, which I think is
because the tongue is much higher than for a normal schwa
(where it is halfway between low and high).
(The French schwa may be closer to the Portuguese one
than those in the other languages, although it may rather
be between front and central, and rounded).
So this European Portuguese schwa is practically a high vowel. It is similar to the Russian sound written with the cyrillic letter which looks like "bI" (ы), and which is often rendered "y" in Latin transcriptions of Russian.
The Handbook of the International Phonetic Association uses the symbol "turned m" (Unicode symbol hex 026F, ɯ) for this sound. So they say it's a high unrounded back vowel, not high unrounded central. There may well be some truth in that. If so, it means the vowel is close to what in Turkish is written with a dot-less i (Unicode symbol 0131, ı).
Further back-up for that theory follows from listening to Daniel Jones's
recordings (dating from 1956!) of
secondary cardinal vowels:
So far my theory was that the Portuguese sound is
cardinal vowel number 17, but it doesn't sound very much like that
in Mr. Jones's recording. It isn't exactly
cardinal vowel no. 16 (which is back unrounded, not central
unrounded as is
either, but it comes close. The truth is probably somewhere in the middle.
Note that in his explanatory text Mr. Jones mentioned cardinal 16 as occuring for the Russian phoneme written bI (in real Cyrillic ы) before a dark l, and cardinal 17 for that same phoneme when before a palatalised t.
In Portugal, the high schwa (as I will continue to call it, for simplicity's sake) changes to a front vowel [i] whenever it is immediately followed by another vowel. It's only a small change, because both [i] and  are high (close) vowels, which differ only in that the highest point of the tongue is in front versus central position. So I thought for a while that this change didn’t really take place, but only seemed to happen, in the ears of foreign listeners familiar with high front sounds but not with high central sounds. But on careful listening, the change is actually very real, as in this example: Só de ouro ©.
Especially from the samples sempre ©, horizonte © and que ©, it is quite clear that the central sound is very different from a schwa in other languages. Arguably, French is an exception to this, but remember that almost-front rounded sounds (French schwa?) may produce a rather similar effect, and so are often confused with, almost-back unrounded sounds (Portuguese schwa).
In connected speech, all Portuguese vowels but especially /i/, /u/ and /1/ are often extremely short in duration. For example querer /k1rer/ (to love/want) and crer /krer/ (to believe) often have practically identical pronounciations. Examples: querer © and querer ©.
It could be said that where many languages have a single "schwa", European Portuguese has two, a high schwa and a low schwa. It is probably a coincidence, but there is a similarity here with Romanian (or use this link in a browser with (limited) Unicode support). The open Romanian schwa is written a-breve (ă), and the close "schwa" is written â in names, and î in normal words. Another striking similarity is that Romanian has a phonemic voiceless vowel /i_O/, and that Portuguese too has (albeit non-phonemic) voiceless vowels.