As mentioned in the comments already, there isn't one universally accepted correct notation. And the prevalence of both notations throughout the literature would make it quite a Herculean task to correct this notation in the writing of others.

I find aspects of both notations bothersome. So in my own writing, I tend to avoid using either. Instead, I write let P have coordinates (1,2) or P has coordinates (1,2) or P is a point with coordinates (1,2).

The problem I have with P=(1,2) is, perhaps, a philosophical one (but being logician by training, I like to think philosophically). To me, it seems that the plane exists prior to the coordinate system. We are perfectly capable of talking about points and their relationships amongst themselves and other objects without taking recourse to any coordinate system. (Read Euclide to see the extent to which he avoides ever even talking about numbers. Indeed, coordinate systems are a rather modern development.) After all, there are other coordinate systems, why identify a point with one particular choice of coordinate system. Consider how, if we decide our plane is inbedded in a larger 3 dimensional space (or even higher dimension), the point is still P, but it's coordinates must change: let's say surreptitiously P has coordinates (1,0,2) in this higher dimension. Given the transitive nature of equality, would we really want to say (1,2)=(1,0,2)? I think not. But all notations have their short-comings: so perhaps in this context we're willing to disclaim that the transitivity of equality can be so cavalierly applied.

A coordinate system helps us locate the point and differentiate one point from another via functions: that is, a coordinate system allows us to analyze what's happening between points by reducing the analysis to that of what's happening between numbers and vectors etc. And a great deal of mathematical thinking has gone in to showing that thinking about numbers and vectors will not lead us astray in regards to points and their ilk.

I would say it's best to look at the context in which a particular notation is being used as respect that within a particular field one style of notation may be preferred over another. Perhaps I overgeneralize, but I don't believe many mathematicians put too much thought into why they use a particular notation until they find the notation creates more problems then it solves. But also, mathematicians are generally not worried about the deeper philosophy a particular notation suggests.

As mentioned in the comments already, there isn't one universally accepted correct notation. And the prevalence of both notations throughout the literature would make it quite a Herculean task to correct this notation in the writing of others.

I find aspects of both notations bothersome. So in my own writing, I tend to avoid using either. Instead, I write let P have coordinates (1,2) or P has coordinates (1,2) or P is a point with coordinates (1,2).

The problem I have with P=(1,2) is, perhaps, a philosophical one (but being logician by training, I like to think philosophically). To me, it seems that the plane exists prior to the coordinate system. We are perfectly capable of talking about points and their relationships amongst themselves and other objects without taking recourse to any coordinate system. (Read Euclide to see the extent to which he avoides ever even talking about numbers. Indeed, coordinate systems are a rather modern development.) After all, there are other coordinate systems, why identify a point with one particular choice of coordinate system. Consider how, if we decide our plane is inbedded in a larger 3 dimensional space (or even higher dimension), the point is still P, but it's coordinates must change: let's say surreptitiously P has coordinates (1,0,2) in this higher dimension. Given the transitive nature of equality, would we really want to say (1,2)=(1,0,2)? I think not. But all notations have their short-comings: so perhaps in this context we're willing to disclaim that the transitivity of equality can be so cavalierly applied.

A coordinate system helps us locate the point and differentiate one point from another via functions: that is, a coordinate system allows us to analyze what's happening between points by reducing the analysis to that of what's happening between numbers and vectors etc. And a great deal of mathematical thinking has gone in to showing that thinking about numbers and vectors will not lead us astray in regards to points and their ilk. But I would caution that there's nothing obvious about how coordinates relate to each other being equivalent to how points relate to each other. And in that sense P=(1,2) suggests that this equivalence properties obvious where, in fact, it took a long time to be discovered and verified.

I would say it's best to look at the context in which a particular notation is being used as respect that within a particular field one style of notation may be preferred over another. Perhaps I overgeneralize, but I don't believe many mathematicians put too much thought into why they use a particular notation until they find the notation creates more problems then it solves. But also, mathematicians are generally not worried about the deeper philosophy a particular notation suggests.