When Jesus therefore saw his mother, and the disciple standing by, whom he loved, he saith unto his mother, Woman, behold thy son! Then saith he to the disciple, Behold thy mother! And from that hour that disciple took her unto his own home.
Yes...this is directed to John.
The Greek behind own indicates this was very private, personal. It is particular to the individual. In this case John.
It does not mean what the catholic wants it to mean that Christ gave Mary to all of us.
Honour thy father and mother; which is the first commandment with promise; That it may be well with thee, and thou mayest live long on the earth.
And I am sure John did this.