Having worked in a marketing department, I can say that having your brand identity become synonymous with a product isn't necessarily a good thing. Let's use the Band-Aid brand as an example.

Once Band-Aid becomes the lingo for every adhesive bandage, then the true Band-Aid brand ones lose their value because it becomes increasingly difficult to make the real ones stand out from the generic ones.

If you look at advertising for brands like Band-Aid that have lost their brand identity, they use the name and its description. This tends to get very wordy and makes the advertising lose impact. You get calls-to-action like the following:

"Use Band-Aid brand adhesive bandages because they're the best!"

instead of:

"Band-Aids are the best!"

Edit: fixing a typo :)