If you wear a shield without proficiency, you have disadvantage and can not cast spells
As you point out, shields are not armor, although the language use is not fully consistent (and this is an issue that has plagued earlier editions, too). For example, shields are listed in the "Armor" table, not the "Armor and Shields" table on p. 145, but the section in the rules is called "Armoer and Shields". Most of the time, armor just means armor, not shields, but to really understand if "armor" in a passage refers to just bona fide armor like plate armor, or is used in a more general sense, meaning "armor and shields", you need to look at the context.
The problem is this passage (p. 144 PHB) seems to indicate that only wearing armor witout proficiency causes disadvantage or blocks you from casting:
Anyone can put on a suit of armor or strap a shield to an arm. Only those proficient in the armor’s use know how to wear it effectively, however. If you wear armor that you lack proficiency with, you have disadvantage on any ability check, saving throw, or attack roll that involves Strength or Dexterity, and you can't cast spells.
The sentence that talks about disadvantage only mentions armor, and omits shields. Does this mean that equipping a shield without proficiency does not impose disadvantage on rolls and does not stop you from casting?
No, it does not. You cannot ignore context or other parts of the rules, to understand if this use of "armor" is meant to include shields.
Context: The first sentence talks about both armor or shields, it mentions "or strap a shield to an arm", setting context that this paragraph is about both armor and shields.
Class description rule text Shield proficiency is listed in the "armor" line of the Proficiencies section, along with any armor proficiency. This is indication that the term "armor" can include shields, when talking about proficiency.
Other Rules text: The PHB says on p. 14: "There are drawbacks to wearing armor or carrying a shield if you lack the required proficiency, as explained in chapter 5." If only armor not shields caused the drawbacks, carring a shield without proficiency would not have a drawback, and this is in direct contradiction to this rule text.
From these it is clear that the section here must mean "armor including shields" when it talks about the effect, and that is what the spellcasting rules mean to, too, when they talk about being unable to cast in armor.