Are people's names always pronounced the same in Mandarin?
Or do all Chinese names have only one possible reading in Mandarin?


No.

Apart from first names (when 多音字 (polyphone) is used they might pronounce differently), even for the last names, there're some (not much) with different pronunciations. Note that these names with different pronunciations almost means different origins. That means you have to confirm with the person about the pronunciation of his/her given name; on the other hand, if you know that you could get more informations, like how the name originated (from where or from who), and the informations about his/her ancestors. e.g.


费（Bì，  Fèi）
覃（Tán， Xún， Qín）
郗（Chī， Xī）
贲（Bēn， Féi）
隗（Kuí， Wěi）
郇（Xún， Huán）
乐（Lè，  Yuè）
召（Shào，Zhào）



References:
这些姓氏你能正确读音吗？（汇编）
「費」姓該怎麼讀？
一个可能被误读的姓 乐姓