No. The Glaaar-pat feature is not described as magical, so it is not magical.
The Sage Advice Compendium tells us how to determine if a feature is magic, in response to the question, "Is the breath weapon of a dragon magical?:
Determining whether a game feature is magical is straightforward. Ask yourself these questions about the feature:
- Is it a magic item?
- Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
- Is it a spell attack?
- Is it fueled by the use of spell slots?
- Does its description say it’s magical?
If your answer to any of those questions is yes, the feature is magical.
Let’s look at a white dragon’s Cold Breath and ask ourselves those questions. First, Cold Breath isn’t a magic item. Second, its description mentions no spell. Third, it’s not a spell attack. Fourth, the word “magical” appears nowhere in its description. Our conclusion: Cold Breath is not considered a magical game effect, even though we know that dragons are amazing, supernatural beings.
Since the song is never described to be magical, is not a magic item, is not fueled by spell slots, does not create the effects of a spell, and does not use a spell attack, it is not magical.
However, it should be noted that the material found in the Sage Advice Compendium consists of official rulings, not rules:
In a typical D&D session, a DM makes numerous rules decisions—some barely noticeable and others quite obvious. Players also interpret the rules, and the whole group keeps the game running. There are times, though, when the design intent of a rule isn’t clear or when one rule seems to contradict another.
Dealing with those situations is where Sage Advice comes in. This column doesn’t replace a DM’s adjudication. Just as the rules do, the column is meant to give DMs, as well as players, tools for tuning the game according to their tastes. The column should also reveal some perspectives that help you see parts of the game in a new light and that aid you in fine-tuning your D&D experience.
If the DM wants to rule that the Bullywug's song is magical, they can. The advantage of the ruling presented for this scenario is that if provides a relatively objective and non-cryptic way to determine if a feature is magical. How do I know a feature is magical? It will tell you.