Welcome to the forums @marblesdev!
The reason behind this may be a little confusing, but I will do my best to explain. In Godot, almost every resource is given an RID (resource ID) when it is used in a scene. This RID is how Godot tracks what node(s) are using what resources. If multiple nodes have the same resource, like the same font, then they all have the same RID. In this case, because the animation you are playing modifies the Font, the Font is a resource with an RID, and all of the instances share the same RID, the AnimationPlayer will affect all of the instances. In other words, the labels all are using the same font file, and so even though there are multiple AnimationPlayers, they are all affecting the same Font resource and therefore whatever AnimationPlayer is processed last is the one that sets the property on the shared Font resource.
There are a couple ways to fix it. The easiest, but least performance friendly, is duplicating the Font resource for each label when it is created. To do this, you would just need to add the following in _ready (untested):
var label_font = get("custom_fonts/font")
label_font = label_font.duplicate()
set("custom_fonts/font", label_font)
However, this will make a new Font resource for every label, which may bog down performance. Instead, what I would recommend doing is seeing if you can find a way to make a similar effect to the inflate animation by changing the Label's properties. You can probably get something similar to the effect by changing the Label's scale. Regardless, changing the Label node's properties would avoid the issue and not negatively impact performance because each node has separate properties that are not shared between instances by default, and these properties are just copied as part of the instancing process (I.E no extra performance cost than just instancing them to begin with).
Hopefully what I wrote about Fonts and RIDs makes sense! Its one of those things that can be confusing, especially in cases like this where you would expect the Font Resource to be unique to each Label, rather than shared across all instances.
