Is it possible to avoid using the article in the following sentence:
I have got a Playstation.
In English you need an article before a noun except when the noun is already preceded by a number or certain other words that identify quantity, or when it is a proper noun, i.e. the name of a unique thing.
So, "I have a dog." (article needed) "I have two dogs." (no article needed -- number) "I have several dogs." (no article -- "several" works like a number) "I have Rover." (no article needed, Rover is the proper name of a specific dog)
The case you give is potentially confusing because "Playstation" may look like it's a proper name. But it isn't, because it doesn't identify a unique object. There are many Playstations out there. So the correct usage is, "I have a Playstation."
If you gave names to each of your video game consoles and you like to call this one "Playstation Zebra" [subtle cinematic allusion for old people], then the correct usage would be, "I am using Playstation Zebra". You would not include an article because Playstation Zebra is a proper name.
Maybe it helps clarify to point out: if you own a Nintendo game console, you would say, "I own a Nintendo", because it's one of many. But if you bought the company, you would say, "I own Nintendo", because there is only one Nintendo company, so it's a proper name.
Side note: The fact that there may be more than one of something with the same proper name doesn't make it not a proper name if you are using it to refer to an individual and the common name is a coincidence. Like if you had a dog named Rover, you would say, "I took Rover for a walk" -- no article -- even though there may be other dogs named Rover. You are not considering your dog to be a member of the class of Rovers, rather, he is an individual who is named Rover.
The marketing teams at Apple/Sony/Nintendo/etc. would dearly like to get everyone to drop the article when referring to their "flagship" products, because it makes those products sound even more important if they can stand alone without "the" or "a" ("the" is considered "better" in this context than "a").
So if you want to sound like a truly dedicated fan of the product (or if you work for the company), then drop the article. Otherwise, leave it in.
"I have a Playstation, but you have an XBox" > "I have P, but you have X", in which the brand is applied not to the object but to the experience of using it. Similar to "Thank you for shopping K-Mart" in place of "--- at K-Mart", in which, like a physical object (the game console), the brand that applies to a physical place (the store) has become the label for a supposedly uniquely satisfying (shopping) experience.