In the 5e D&D core rules "shaman" and its derivatives don’t appear in the PHB. It’s used in the MM to refer to stone giants, lizardfolk, quaggoths, and NPC druids. In the DMG it refers to lizardfolk shamans.
The PHB PC druid is a "priest of the Old Faith" whereas the MM NPC druid is a "tribal shaman". "Old Faith" refers to pre-Christian European magico-religious beliefs and practices. "Shaman" in popular culture is associated with non-European, particularly Native American, magico-religious beliefs and practices.
All four MM shamans provide services to their people. Stone giant shamans "draw shapes out of raw stone, which they believe reveal meaning inspired by their god, Skoraeus Stonebones." "Lizardfolk shamans lead their tribes, overseeing rites and ceremonies performed to honor Semuanya." Quaggoth tribal shamans are also known as thonots. "A thonot keep [sic] a tribe's lore and ensures its superiority against enemies." NPC druids "heal the sick, pray to animal spirits, and provide spiritual guidance".
Lizardfolk and quaggoth religion involves evil acts. Humanoid prisoners "are sacrificed to Semuanya, the lizardfolk god". "A thonot that fails the tribe is slain and devoured in a cannibalistic ritual, in the hope that its power passes to another more worthy quaggoth". There is a suggestion of superstition about stone giant and quaggoth religion – "believe", "hope".
Stone giants, lizardfolk, and quaggoths, despite their alignments (neutral, neutral, chaotic neutral) are likely to have a hostile relationship with PCs, based on other parts of their entries in the MM:
"When trespassers stray too far into the mountain territory of a stone giant clan, those guardians greet them with hurled rocks and showers of splintered stone."
Lizardfolk are "territorial xenophobes". "When unwelcome visitors are detected, a tribe sends a hunting band to harass or drive the trespassers off, or tricks them into blundering into the lairs of crocodiles and other dangerous creatures." "Any creature that enters their territory is fair game to be stalked, killed, and devoured. They make no distinction between humanoids, beasts, and monsters."
"Savage and territorial, quaggoths climb the chasms of the Underdark. They maul their foes in a frenzy, becoming even more murderous in the face of death."
All three, imo, ought to be of evil alignment by the PHB definition. "Neutral evil (NE) is the alignment of those who do whatever they can get away with, without compassion or qualms." There is some strange text in the stone giant and lizardfolk entries that uses, respectively, false beliefs and cultural relativism to justify their neutral alignments:
Stone giants view the world outside their underground homes as a realm of dreams where nothing is entirely true or real. They behave in the surface world the way humanoids might behave in their own dreams, making little account for their actions and never fully trusting what they see or hear. A promise made above ground need not be kept. Insults can be made without apology. Killing prey or sentient beings is no cause for guilt in the dreaming world beneath the sky.
"Lizardfolk have no notion of traditional morality, and they find the concepts of good and evil utterly alien. Truly neutral creatures, they kill when it is expedient and do whatever it takes to survive."
Sacrifice of sentient beings is going beyond "whatever it takes to survive". If cultural relativism applies to lizardfolk then why not to other monsters in the MM? Don’t behir (NE), ettercaps (NE), grell (NE), kobolds (LE), perytons (CE), and yeti (CE) also "kill when it is expedient and do whatever it takes to survive"?
The stated alignments for stone giants, lizardfolk, and quaggoths may explain why the writers felt they could give them shamans.
Even if we accept the MM alignments, shamanism is being associated with cultural beliefs - the surface world is a dream, rejection of the "concepts of good and evil" - that permit actions most of us would consider to be immoral.