Glucose is sugar, but not all sugar is glucose - "sugar” is a category and glucose is one member. Others sugars are fructose, galactose, sucrose (aka table sugar = glucose + fructose), and lactose (milk sugar = glucose + galactose).
Glucose is a simple sugar (a monosaccharide): C₆H₁₂O₆. One of the most basic carbohydrate molecules. It's form of sugar that your bloodstream carries and your cells use for energy. In blood tests, when people say “blood sugar”, they are referring specifically to blood glucose. Starch consists of chains of glucose molecules that break down into individual glucose molecules.
Generally when people talk about dietary sugar they are referring to table sugar. The fructose part of table sugar doesn't raise blood glucose directly and doesn't trigger much insulin and isn't used by your cells for energy directly. It travels straight to the liver. The liver converts fructose mostly into fat as well as glucose, glycogen, and lactate. Larger amounts of fructose contribute to getting a fatty liver - extremely bad for your health.
When people talk about a no sugar diet (or very low sugar diet) the sugar they are talking about is table sugar - they usually do not mean a glucose free diet. The major health benefit of restricting table sugar is that it reducing fructose exposure to your liver, with a side benefit of not spiking your blood glucose levels with the glucose.
You are correct that your body needs glucose: red blood cells, parts of the brain, kidneys, and the retina all need glucose. So how do people who are fasting for multiple days or who don't eat any carbs survive? Fortunately your body can make glucose with a metabolic process called gluconeogenesis (“making new glucose”) that happens mostly in the liver (and a bit in the kidneys) from non-carbohydrate sources - mostly protein and fat (from food or your own body if you're fasting).
For people who consume no carbs or very small amounts of carbs, the parts of your body that don't absolutely require glucose switch to fat and ketones. The body adjusts with most organs burning fatty acids and the brain starts running on ketones, reducing glucose requirements by about half. This spares muscle from being used for glucogenesis, because the brain no longer needs 120 g/day of glucose — it might need only ~40–60 g/day.
Of course, someone with type 1 diabetes needs to manage their insulin use carefully if they reduce their carb/sugar intake to avoid their blood glucose going too low. There are glucose only (no fructose) products that you can use to raise your blood glucose - tablets, gels, powders, and drink mixes. Often called “Dextrose”, which is just the food/medical industry name for pure glucose.