Why do we say that the model has a high variance when variance is actually the measure of spread of the data and not some property of the model? I am trying to understand the difference between bias-variance and overfitting-underfitting. If a modal overfits the data it means that it will not generalize well on new data because it over learns the training dataset. I can also say that this overfitting model is biased for the training examples. Now the question is, what is variance? Why does my overfitting modal has high variance when variance is not a model's property.
P.S. If I become able to make sense of the variance in terms of the model, I will be able to get bias in terms of the model as well.

The other answers makes a great illustration, thus I want to answer your "... when variance is not a model's property."
It can be a bit confusing since a model is not "data" as such, but a "thing" that produces data. But, since a model is used to produce data, when we talk about a "model having big variance" we actually, losely speaking, mean that "the data the model produces have a high variance".
Consider the following; say you have a model which always predicts the number 5 e.g
X = [1,2,3,4,5,6,7]
model(X) #[5,5,5,5,5,5,5]

No matter what data you use put into the model it will always output $5$, thus we would say that your model does not vary at all (eventhough it is actually the output that doesn't vary, but since the output is a function of the model, then the model doesn't vary at all either).
Now, say you have a model which does the exact opposite, it varies a lot
X = [1,2,3,4,5,6,7]
model(X) #[-12,0,30000,9,-10000,2500,13]

as you can see (relatively) small changes in your input data results in huge difference in your ouput data (the model has a big variance). With a good model, we would expect that inputs that are close to eachother would result in outputs that are close to eachother aswell, which is not the case here.
These two scenarios corresponds to underfit/overfit respectively (see @Álvaro Méndez Civieta great plots), where the overfitted-model jumps a lot (high variance in the output) and the underfitted model doesn't move at all (or very little).
