Is it possible to display the names of geographic characteristics in a larger fontsize?
Yes, but you will need to build your own rendered tiles. See https://switch2osm.org/.
It is also most likely that you will need to make significant revisions to the standard Carto-CSS style as many of the rules will have been optimised for the font sizes chosen.
In other words: it is possible, but a far larger task than one might expect.
To add to the previous answer, I think you are making the common mistake that the map is image that you see on the main website. That is just provided as a convenience to mappers. The actual map is the underlying data, and that has no concept of fonts or font sizes.
The way that the web site works is not to create the image of the map on the fly, but rather to select appropriate images from those already created to display. As it is relying on pre-created images, it can only display them with the font sizes with which the images were created. It already takes so much processing power to maintain images with one font size that the standard map can be several hours behind, at busy times, and some of the alternatives take up to a week to update. Creating images with different font sizes is not something that the project can afford to do, given the intended purpose of the web site and the limited sources of money to run it.
In general, if you want to create a printable, or viewable, map that is different from one of the standard versions, or you want to make extensive use of such maps for purposes other than maintaining the underlying data that describes the map, you need to run the software that creates the map images on your own machines, and reconfigure it to meet your needs. As you are dealing with technical tools, not consumer ones, you will need to put in a significant amount of effort to learn how to do this, and you will have to configure the tools in much more detail than a normal member of the public would want to do.
(There is one compromise technique which is, or was, used to allow maps to be created in a user choice of language. This is to generate the text on the fly, and superimpose it, possibly in the browser, on a background with no text. However, this is not required to meet the purposes of the main site, and the positioning of the text may obscure features in the base image.)