Q:

Term when a brand name become synonymous of the product it produces

For example most of the people hearing "I really like my BMW" will understand than he is referring to a car whereas someone saying "I do enjoy my Lacoste" will leave doubts about what kind of product(a shirt, a fragrance, etc) he is talking about.
The term where saying 

I  really like my [brand name].

has the same meaning as 

I really like my [product name]. 

For example 

I really like my BMW. 

means the same as (because BMW is so famous that is implicit that it is a car)

I really like my car. 

The term would be fit in:
 BMW is an example of _____, the manufacturer of cars is so well-know that we all know pretty sure that we are talking about a luxury car when we hear "I really like my BMW"

A:

I think the expression you are looking for is proprietary eponym: 
An eponym:

is someone or something whose name is or is thought to be the source of something's name (such as a city, country, era, or product); alternately it can be used to refer to the name of something that is based on or derived from someone or something else's name. Albert Einstein is the eponym of the element einsteinium; conversely, einsteinium is an eponym of Albert Einstein.

Proprietary eponyms

are general words that are, or were at one time, proprietary brand names or service marks. Kleenex, for example, is a brand of facial tissues, yet the word is used today to refer to facial tissues of any brand. Xerox is a brand of photocopy machine, that word, too, has been since adopted to refer to any brand of photocopy machine and, moreover, also employed as a verb to describe the act of photocopying.

(www.rinkworks.com)

Active Trademarks Often Used Generically. 

