Other parsers (e.g. ETE Toolkit) don't have issues with single quotes in the node names.

Single quotes do not work on ETE either. Or rather, it appears to work but isn't interpreted correctly. It just doesn't fail. At least that is the case for their website in that link as of today.

UPDATE: silly example ('flower (Linnaeus, 1707:1778)':3, shrub:2)plant:0;
to the best of my knowledge anything is allowed inside single quotes (expect single quotes that need to be escaped by another single quote).

Have you checked your file is valid according to the specification? The details are not fresh in my mind, but perhaps a single quote in a name needs to be escaped to be valid?

I've hit a superficially similar issue related to single quotes. They don't round trip correctly.

A reproducible example is very simple, doesn't require loading a huge genome:

from Bio import Phylo

tree = Phylo.BaseTree.Tree(root=Phylo.BaseTree.Clade(name="Root"))
clade_with_quote = Phylo.BaseTree.Clade(name="Node'Name")
tree.root.clades.append(clade_with_quote)

with open("tree_with_quote.newick", "w") as file:
    Phylo.write(tree, file, "newick")

with open("tree_with_quote.newick", "r") as file:
    tree_read = Phylo.read(file, "newick")

original_node_names = [clade.name for clade in tree.find_clades()]
read_node_names = [clade.name for clade in tree_read.find_clades()]

mismatches = [(original, read) for original, read in zip(original_node_names, read_node_names) if original != read]
print(f"Mismatches: {mismatches}")
# Mismatches: [("Node'Name", "Node\\'Name")]

I've opened two separate issues related to erroneous serialization and deserialization in #4536 and #4537