It depends of the underlying geometry you are using.

99.9999% of time it so called “euclidean” geometry 1 plus 1 means according to Euclide book “the elements” : what is the size of a segment of size one to which I add another segment of same size at the end?

2.

Euclide was a teacher learning most notably with a stick and making drawings on the grounds.

What if 1 was a speed? And this speed was closed to c (the speed of light)? 1 + 1 would be combining both speed ….

1 + 1 would be 1 because we would be in Einstein’s relativity… because we are projecting an hyperbole on a plane seen from the top (that is the einstein relativity).

As Euclide said the meaning of mathematical symbols is not absolute, it is relative to a context and the art of measuring stuff and doing simple operations.
+ is be concatenation or translation (where am I if I am at position 1 and move from 1)
x (multiply) homothetia (the art of elongating stuff)
….

This context leads to geometry, and science (not math) is the art to figure out the way to measure (métros) the world/earth (geo) hence theoritical science is the art of devising in which geometries we live according to contexts and will always automatically provide predictions about what happens when we will measure 1 + 1. Experimentation is the art of measuring and checking that theory actually works.
Engineering is the art of puting this into practice.
Mathematicians are like religious, they think they can see truth no one can in a simple stuff and are lost to the world, never trust them.

There are no limits to the geometry you can make. “The elements” of Euclide is probably the best book to learn, understand and teach what 1 + 1 is. And actually this book gives you also the recipie to build your own geometry.