TL;DR: Yes. Essentially it's "put away $X per year for Y years, and you will have $Z when you retire", where the source of X doesn't matter (your money or your employer's match).
If you look at JTP's answer to your linked question, you'll see that the 15% recommended there is "10% saved and 5% matched". Other people who recommend this savings rate may assume a different breakdown, or may assume no match at all. It really depends on the assumptions and calculations used by a particular person to arrive at that recommendation.
That question and its answers highlight that there are a lot of underlying assumptions involved in arriving at the "save 15% for retirement" advice. The question does a good job of enumerating the many variables that are involved in such calculations, but the baseline assumption behind these calculations, and this type of advice in general, is
I am going to work a job for money until I am X years old, and then I will no longer work. The money I save while working should replace Y% of my income until from the time I am X years old until I die.
There are really two high-level steps to calculate this savings rate. The first is "how much money do I need to have saved/invested in order to generate that income?"; this will be an actual number, not a percentage. For example, if you want $50,000 of income in retirement (assuming this is Y% of your current income), you likely want about $1,250,000 saved/invested (or more, or less, depending on the assumptions made about rate of return, withdrawal rate, life expectancy, other income, etc.).
From there, the second step is "how much do I need to add to my savings/investments per year to reach that number when I am X years old?"; this number will also be an actual number. For example, if you are currently X-1 years old with 0 savings, you will need to contribute $1,250,000 per year to reach your goal. If you are X-40 years, with some savings, that number will be considerably less.
The number calculated in the second step may indeed be around 15% of your income. However, it doesn't really matter what the source of the money is, as long as that amount of money gets saved/invested.
This advice generally assumes that a person spends a large portion of their income, and therefore needs to replace most of their work-generated income in retirement. If you instead look at what your expenses are and aim to cover those, rather than replace income (you may get promotions and/or large raises, but this doesn't mean you will need more income in retirement and are suddenly "behind" on saving even though you were "on track" before the promotion), you may find that the savings goal (e.g. $1,250,000) is more than you need. This means you could either set a lower goal and save at a slower rate, reaching your lower goal by age X (although then you may be spending more of your money, making your lower goal inaccurate), or set a lower goal and reach it before age X, allowing you to retire earlier (or continue saving and retire more securely).