B"H
"Are the words Shema and shemiras / shomer related?"
Everything is related to everything in some way.
If you're asking if the words listen (שמע) and guard (שמר) share any unique grammatical similarities that neither share with any other word, then no. They share the first two letters, but are completely different words with completely different roots.
Now, everything is related to everything else, all the more so in the Torah and the Hebrew language. Sometimes two words that are not grammatically similar do have other kinds of similarities.
For example, the words "leven" (חמץ) and "unleavened bread" (matzah/מצה) almost share the same letters, and Torah Ohr from the Alter Rebbe explains that the ח can be interchanged with a ה in some cases, thus highlighting that even though chametz and matzah are total opposites, they still can have some similarities, as explained there (parshas bo).
Anyways maybe the ר and the ע could be similar in some way? But I haven't seen any sources to support this.
Anyways, in general listening to something means one guards it in one's mind, so it does seem the words have similar implications, but no grammatical similarities.