As a visitor to the UK, you are (almost) always admitted for 6 months, regardless of the length of stay which you apply for, or state at the border. However, it is usually a bad idea to drastically extend your stay beyond what you originally asked for. On future application for visits, you could face denial or tough questioning, which is perhaps why you're worried.
However, none of this matters for family visas. For a family visa, even periods of overstay (and you have not overstayed) are disregarded. Generally, refusals following a longer than asked for visit will fall under a failure to prove a genuine intention to visit, specifically paragraph 4.2(a) "The applicant [...] will leave the UK at the end of their visit", since if you stayed longer than you said once, they will likely believe that you will do it again.
But for a family visa, the grounds under which the entry clearance officer can deny you are much narrower. For a refusal on the basis of previous immigration run ins for a family visa, the applicant must have "aggravating circumstances, such as absconding, not meeting temporary admission/reporting restrictions or bail conditions, using an assumed identity or multiple identities, switching nationality, making frivolous applications or not complying with the re-documentation process". Obviously, you haven't done any of this.
I imagine the three months that you spent in the UK will form a significant part of the story of your relationship with your partner, so it will be natural to explain it in the covering letter, and will in fact help demonstrate that you have a genuine relationship. And obviously, you must include this stay in the UK when disclosing your travel history on the application.
So, to reiterate, even if you had a previous overstay, it would not affect your family visa application. Staying longer in the UK than you had previously told UKVI that you would, within the limits of the leave you were granted does not even count as an overstay and will be of no concern to an ECO as long as all stays in the UK are properly disclosed in the application.