Is it correct to say:
This is Bangkok Metro.
This is London Underground.
This is New York Subway.
(Without "the" and without possessive 's)
Is it correct to say:
This is Bangkok Metro.
This is London Underground.
This is New York Subway.
(Without "the" and without possessive 's)
It depends somewhat on what the official name of the subway is.
That said, if one is comparing London's subway with New York city's, then it's reasonable to use lower case as well as possessive form.
Public transportation systems like the London Underground, and New York Subway generally require either "the" or a possessive form.
In informal speech, people may drop "the." Also in titles and logo. Google Ngram shows that the article-less version "London Underground" is more common.
Edit - 1.
But "Bangkok Metro" does not use "the" in its official name.
BMR has two types of metros; namely, Bangkok Mass Transit System (BTS-Skytrain) and Mass Rapid Transit (MRT-Subway), which are located at the heart of the city adjacent to business centers. Running north–south of the inner city, the BTS has two main lines and the MRT has one line; covering distances totaling over 45 km.
On Wikipedia, Indian metro system names appear without "the" in titles, but "the" is used in sentences.
Kolkata Metro, Delhi Metro, Kochi Metro..
Rather than calling it "wrong grammar," you can consider it a stylistic choice.