Is the law of proportions the same as the law of constant composition?

It depends on which law of proportions you are talking about. There is the law of definite proportions and there is the law of multiple proportions. The law of definite proportions is most like the law of constant composition.
First, the law of multiple proportions means that 2 or more can combine in different proportions, i.e. Carbon and Oxygen can produce CO or ##CO_2##.
Whereas, the law of definite proportions, all in a sample need to have the same ratio by mass. For example, CO (carbon monoxide) has a formula mass of about 28 grams per mole, which is made up of 12 grams from the Carbon and 16 grams from the Oxygen. The fraction of Carbon to CO, when reduced from 12/28 is 3/7. The fraction of Oxygen to CO, when reduced from 16/28 is 4/7. These fractions for a specific compound are definite AND constant. Which takes us to the next law.
The law of constant composition is essentially the same idea as the law of definite proportions. It’s defined as all the samples of a specific compound will have the same ratio of elements. (It does not mention by mass, however, that idea is implied.) If you have Carbon Monoxide, you will always have a 1 to 1 ratio of Carbon to Oxygen.