You can do that at least on my MacBook Pro (Catalina 10.15.5).

Open Skype app. Go to menu bar at the left upper corner. Navigate to Preferences -> Messaging -> Text size. Currently, there are five options: Normal, Smaller, Small, Large and Larger.

For those of you that looking for a visual answer

step 1

step 2

hope this help ...

At this moment in the future which is already here and which brought us version 6.12 the solution is:

1. Click the the chat window
2. From the menu select View > Make Text Bigger ⌘+.

There is no way at the moment. In fact there is even a thread in the Idea Garage on Skype's forum about the matter

If an extra click is no too big of a deal, you could make a small script to change it for you that you run each time you run Skype.

Or, if you are adventurous you can try finding where in Skype's code the default font size is set and change it. This may or may not work so please do at your own risk.

I am sorry I couldn't provide a simple solution, I hope this helps.