A 'badge', typically, is something that can be attached to something else - for example, the badge on the front of a vehicle which displays the manufacturer's emblem is often a removable/replaceable item and 'debadging' is a common cosmetic modification to cars.
In the UK, where school uniform is the norm, schools may display their logo (which could also be described as an 'emblem', a 'crest' or sometimes a 'shield', depending on the form it takes) on their uniform. On the blazers/jackets worn by British senior school children, this is typically in the form of a badge (or 'patch' for cloth badges) that is sewn onto the uniform either as part of the manufacturing process or supplied separately to be attached.
What appears in your image is certainly a school logo, but it is not in the form of a badge that has been attached to the garment - it appears to be screen-printed onto it. It could certainly be described as a school 'logo' or 'emblem' but probably not a 'badge'. The shape of the logo is not a 'crest' or 'shield' either.
Note that not all English-speaking countries have school uniforms, so beware of opinion-based answers.