There are a few performance tricks we can apply here:

• add __slots__ to the class definition should help with memory and performance as well:

   class _RegExLib:
       """Set up regular expressions"""
       # use https://regexper.com to visualise these if required
       _reg_school = re.compile(r'School = (.*)\n')
       _reg_grade = re.compile(r'Grade = (.*)\n')
       _reg_name_score = re.compile(r'(Name|Score)')
  
       __slots__ = ['school', 'grade', 'name_score']
  
       def __init__(self, line):
           # check whether line has a positive match with all of the regular expressions
           self.school = self._reg_school.match(line)
           self.grade = self._reg_grade.match(line)
           self.name_score = self._reg_name_score.search(line)
  

• using next() instead of .readline() should be faster as it uses a lookahead buffer internally:

   with open(filepath, 'r') as file:
       line = next(file)
       while line:
           reg_match = _RegExLib(line)
  
           if reg_match.school:
               school = reg_match.school.group(1)
  
           if reg_match.grade:
               grade = reg_match.grade.group(1)
               grade = int(grade)
  
           if reg_match.name_score:
               value_type = reg_match.name_score.group(1)
               line = next(file, None)
               while line and line.strip():
                   number, value = line.strip().split(',')
                   value = value.strip()
                   dict_of_data = {
                       'School': school,
                       'Grade': grade,
                       'Student number': number,
                       value_type: value
                   }
                   data.append(dict_of_data)
                   line = next(file, None)
  
           line = next(file, None)
  

Some of the code style and other notes:

• file is a reserved keyword, think of a different variable name
• define your regular expression strings as raw strings
• you can probably replace .* wildcard with a more concrete \d+ for the "grade" regex: Grade = (\d+)\n