New shoots are sprouting from under the soil just beside my rose plant. At first I thought they were new plants, but now I think they are shoots/branches of the existing plant only. Is this alright?

to the forum Sohamsekhar!
Your rose looks like a cutting or seed grown plant so it's all the same plant. Roses will often send up new stems from the base or even underground. They are related to raspberries and blackberries and grow roughly the same way. It's just your plant showing how well it's doing!

Yes, excellent sign that your plant is growing and thriving!