Why do we not have a true upper limit for dissimilarity measure? The similarity measure between two attributes always falls in the range of $[0,1]$, why is this not true in the case of dissimilarity where the value falls in the range of $[0, \infty)$?
Can't the dissimilarity between them be thought of as the opposite of similarity in the sense that if they have a similarity of 0, then the dissimilarity will be 1 and vice versa?

You are correct that dissimilarity is the opposite of similarity. However, it is not true that all similarity metrics are bounded between 0 and 1. For example, you can use dot product as a measure of similarity and it's non-negative without an upper bound. Cosine similarity is bounded between -1 and 1.
Also, distance (dissimilarity) metrics aren't necessarily unbounded: there are also normalized metrics that have upper bounds.
For many unbounded metrics, when applying them we know what are the practical bounds (it's practically impossible to be higher than something). So both can be bounded and unbounded. The mathematical definition of distance mentions that it's non-negative not to put unnecessary constraints on it, but this doesn't mean that there do not exist distance metrics that are bounded.
Also, keep in mind that having the bound doesn't change much. If you have a metric that theoretically falls between 0 and 1 but 99% of the time the values fall between 0.1 and 0.15, then the value of 0.2 could actually be interpreted as high. The upper bound doesn't necessarily help to interpret the values, with the exception of extremes.
