My Walmart

Technically he is right. but everyone understands what you mean so hes just being too precise.
A correct way would be: The Walmart i usually go to...

I vote for "my __" as being grammatically correct; it's colloquially common and universally understood through context. (Nobody thinks you are Walmart Corp., for example.) The complainant is a psuedo-grammarian.

What you mean is "my nearest Walmart", "my favorite Walmart" or similar.

You can tell Walmart near my house for example

The reader is not right, not even technically. If I talk about "my teacher," do I really need to clarify that the person is not a slave?

I accept the use of "My Walmart" to mean the Walmart closest to me because: (i) I myself use it that way; and (ii) I often hear others use it in that way. And, I do believe it is prudish to object to this usage.

I would add one more distinction: "My Walmart" can have at least three meanings that I can think of:
(a) the Walmart closest to my home;
(b) the Walmart closest to my work - as I myself often go from work to other places; or
(c) the Walmart I visit regularly, regardless of its proximity to my home/work, as I often visit friends out-of-state and I enjoy going to the Walmart near them.

I accept the use of "My Walmart" to mean the Walmart closest to my home, and I also think it's prudish to object to that usage.

However, I would also point out that "My Walmart" can have at least four possible meanings:
1. that Walmart closest to my home;
2. that Walmart closest to my work (I often leave from work to go elsewhere);
3. that Walmart that I visit regularly, regardless of its proximity to my home/work; or,
4. that Walmart that I prefer, regardless of its proximity to my home/work.
(I often visit an out-of-state Walmart as I enjoy it and I visit it when I am visiting my out-of-state friends.)