As @Scimonster mentioned in the comment, this has to do with the ambiguous definition of the word 'vegetarian'. Depending on who you ask it can either mean 'does not eat meat or fish' (as in ovo-lacto-vegetarian) or 'does not eat animal products' (as in vegan). Both are valid definitions, so calling eggs vegetarian is not wrong.
To back this up, here are some definitions of the word 'vegetarian':
Merriam-Webster (I refer to the definition of the adjective, but the one for the noun is similar, number 2 is the one of interest here)
1 : of, relating to, or suitable for vegetarians
: not containing meat : consisting wholly of vegetables, fruits, grains, nuts, and sometimes eggs or dairy products
<a vegetarian diet> <vegetarian lunch>
Oxford Dictionary (This one relates to the noun. number 1 is the one of interest here)
1 : a person who does not eat meat : someone whose diet consists wholly of vegetables, fruits, grains, nuts, and sometimes eggs or dairy products
2 : herbivore
A vegetarian is someone who lives on a diet of grains, pulses, legumes, nuts, seeds, vegetables, fruits, fungi, algae, yeast and/or some other non-animal-based foods (e.g. salt) with, or without, dairy products, honey and/or eggs. A vegetarian does not eat foods that consist of, or have been produced with the aid of products consisting of or created from, any part of the body of a living or dead animal. This includes meat, poultry, fish, shellfish*, insects, by-products of slaughter** or any food made with processing aids created from these.
As you can see all of these state allow for either variation, and the list is far from complete, if you look up the definition for vegetarian, you will mostly find that it has this ambiguity.