In general, yes.
But in this particular case, “Oil of Olay” is a brand name. They have many products that aren’t based around oil.

I don’t understand the issue.

Oh. OIL FREE OIL of Olay.

Yeah, it’s a brand, Dairy Queen doesn’t change it’s name when you buy a hot dog.

It’s the same as all the PINK stuff that’s just black and white.

It used to be called Oil of Ulay here in the UK. When I went to Australia in the late 90s it was Oil of Ulan. Then one day I discovered it was Oil of Olay. I thought I was losing my mind and that this was a classic case of the Mandela Effect. Turns out it was just marketing, like Marathon becoming Snickers.