When you are using the following call:
.where('name', isGreaterThanOrEqualTo: customer)
It means that you are searching the database for the documents in which the "name" property holds a value that is greater than the one you are typing. In this particular case, both "Omar" and "Second Omar" are present in the result-set, since each one of them starts with a letter that is alphabetically ordered after "k". In fact, all capital letters are present in alphabetical order after the lower-case letters. Remember that the search is performed lexicographically.
If you are looking for a full-text search, please note that Firestore doesn't provide one. As @Dharmaraj mentioned in his comment, you should use a third-party solution. You can also find more info in my answer from the following posts: