When fighting against multiple of the Giant Spider creature and more than one manage to web you, do you need to break free from each web separately, or does just one successful escape check break you free from all webs?
You need multiple checks to escape.
This is a byproduct of the "Combining Game Effects" rule:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items. See the related rule in the “Combining Magical Effects” section of chapter 10 in the Player’s Handbook.
Because of this rule, we can only be affected by a single web at a time, so we can only make a check against one web at a time. Once the effect of one web ends, we are immediately subject to the effect of the second web, as their durations no longer overlap, so a second check is necessary.