Weight loss is definitely an aspect of fitness, and strength training/focus is not all there is to fitness.
No you do not need to track strength while losing weight or even be concerned about it, if that is not your goal. You can lose weight without lifting.
I however suggest that people do allow some kind of strength training while losing weight. Number of reason. Helps burn more calories, helps maintain muscle mass while losing weight. It also provides something to keep you motivated when the scale doesn't budge. You can sometimes go weeks without dropping a pound, but still see an increase in strength, inches lost, etc.
I'm not saying go out and do what I did and become a bodybuilder or anything. I'm just saying it's a bit more extra work for another rewarding experience.
Lastly I saw someone mention something about p90xers vs bodybuilders. This is an apples to oranges comparison. Bodybuilders aim for size and aesthetics foremost, strength and functionality are similarily important but the routines are often less geared towards it.
There are also bodybuilders who do maintain a focus on things like flexability and power. Hell, I even try to keep a focus on this as well.
You train for what you want to do. In my eyes, a runner is no more or less fit than a bodybuilder, powerlifter, speed reader, etc. You train your body for different things. There really isn't a wrong path.