No, multiplayer games have existed for years and years before the term GaaS was invented and these games had long post-launch support, I'll use Warhawk (PS3) as an example. The game remained popular for YEARS after release receiving free updates in addition to paid DLC. It was never called a GaaS, the term wasn't even invented.
The one thing that pretty much all GaaS have in common is the monetization aspect. If you look at the console/PC space all the popular GaaS are multiplayer focused (as opposed to mobile where you have stuff like Pokémon Go that are more solo endeavours). The one thing I appreciate about Helldivers is that I can enjoy it as a single player experience (I've clocked 50h total about 30h of which were solo play). Even when playing solo though you're still connected to the online world, it was fundamentally designed to be a multiplayer title.
As I said before, multiplayer games and GaaS are not the same thing.
Multiplayer games means multiple players can play the game. Single player games is when only one player can play it.
Game as a product is the traditional old school business model approach of shipping a game and moving on, maybe after launch making only a few paid DLCs, that maybe weren't even considered before releasing the game. They can be both single player or multiplayer.
Game as a Service, or live service games, means they are designed to be run during a long period of time getting new content or features, plus tweaks and fixes, a service that evolves over time depending on performance and feedback. Meaning, to have a meaningful post launch support. GaaS can be both single player or multiplayer. I'd say live service games started in the early '00s with the first MMOs and casual browser games (most of them SP games) that later evolved into mobile games.
Live service games often focus the monetization not in the sale of the game, but instead in post launch stuff, which can be of many types: micro transactions, dlcs/expansions, season or battle passes, subscriptions, or even in recent times via blockchain. But there are some who rely more (or only) in the game sales plus paid expansions/big dlc sales.