No, that blood tester is a blood glucose meter. Although it is public, the test strips used to test blood sugar are disposable. A new one is used every time, and it touches and absorbs blood. The site is a test strip, so it is not contagious. The blood collection needles used by nurses to collect blood are also disposable and will not be contagious as long as they are qualified.
In addition, hospitals usually measure blood glucose by drawing venous blood. Why does the hospital you go to still use this portable blood glucose meter to measure finger blood glucose?