It can go either way! Normally, it's meant to be romantic, but it doesn't have to be. It's all based on the tone and the context. For example, "bromances" are quite common, and men tend to use romantic language to express how close they are as friends (in a partly-joking way). It can also be used in a work environment if you completely trust someone to complete a task. It can even be used to flirt - not letting someone know whether you mean it in a friendly or romantic way, but leaving it a little bit mysterious.
In general, if you use "my guy" or "my girl," you are expressing either some strong feelings or a lot of trust.
In some cases, people might be slightly offended if you're being completely serious and romantic (because it sounds as if you're saying you own the other person), so I would recommend using it in a more joking way.