In the modern day, the phrase "any other" is often used to refer to something that is not important or significant. For example, someone might say that they don't care about "any other" sports team besides their own. This use of the phrase is a way of emphasizing that something is not worth considering.
The allusion to the biblical phrase "any other gods before me" is a reminder that there are some things that are more important than others. It is a warning against putting anything before God, and it is a reminder that there is only one thing that is truly worthy of our worship.