The history of secretarial work is very interesting.  You may wish to note that, originally, secretaries were men!  It was the advent of the typewriter that turned it into a profession. If you look at these historical characters, male secretaries usually worked for aristocratic employers, dealing with confidential personal business and the management of estates and personnel.  When the typewriter came along, it was considered very suitable for women because the keyboard resembled the keyboard of a piano!  

I didn't set out in my professional work to make administration my career.  The reasons why I eventually chose it are private, and I was a  late starter in this field, but I can honestly say that I have found it very rewarding, both in terms of job satisfaction and remuneration package.  Apart from my very first job, I have always worked at the top of an organization and have had some extraordinary jobs working for high net-worth and socially prominent people, managing substantial budgets and teams of people.  Those jobs were very hard work, but equally great fun.  My current post is a walk in the park compared to some of those.

Is administration a career? Well, that very much depends upon your outlook, education and training.  If you want a job in which you work at a high level and you have to make important decisions, then those jobs are definitely out there, but you have to equip yourself to do these.  A thorough grounding in office procedures and technical skills as well as good business acumen and, possibly nowadays, a business administration degree will stand you in very good stead.  The label on the package (Admin Assistant, Executive Assistant, Personal Assistant etc) doesn't matter.  What does matter is what decisions you are authorized to make, what budgets you control, people you manage and your remuneration package.    

The word "career" implies a ladder of some sort.  Most people in any sphere of work do not have "careers" they have jobs.  Developing a real career requires thought and dedication - personal investment of time and money in education and training.