Not necessarily always "the". In all of these examples, either the definite or indefinite article is possible. "the" would indicate there was only one, while "A/An" would suggest there was more than one. So (as always!) the speaker would decide what s/he wanted to mean. Eg if the speaker knew that Alcácer was the only tax collection center in the valley, then
Until 844, Alcácer functioned as the center for the collection of taxes in the lower Sado valley But if s/he knew there were several centres, of which Alcácer was just one, then
Until 844, Alcácer functioned as a center for the collection of taxes in the lower Sado valley The same with all the others.
Enjoy the candy
