Disclaimer: This is probably not about your situation. But it might be useful for future readers based on the question title.
You're not saying where you are, but there are situations where this is useful. Let's say you are in Germany, you're from a non-EU country and you hold Blue Card visa (§ 19a). This visa is tied to your job title, but not the company you work for.
Every time you change your job, you need to notify the Auslaenderbehoerde (foreigner's office). If you only change the company, but the title is the same, you just go there, give them a copy of your documents, and that's it it. But if your title changes, they have to approve the job change. You need to prove to them that you still qualify for this kind of visa, and they have to reissue a part of your visa. That part is an additional official piece of paper that says what what kind of job you are allowed to hold.
Examples of changes that need to be approved are Software Developer to Senior Software Developer or Software Engineer to Software Developer, or Technical Project Manager to Project Manager, or vise versa. Obviously Software Engineer to Web Developer also would have to be approved.
Because getting the visa changed is a huge hassle and in cities like Berlin entails about one day of wasting away in the overcrowded Auslaenderbehoerde, it's very normal to ask them to give you a slightly different title. It means you can start earlier, and you're spared the ordeal of dealing with the authorities yet again. For expats in Germany that's a huge thing.
So it's totally legit to ask for that. But you'd ask after you received an offer with a work contract to look at. Making that change then shows you're committed to them. Unless it's a huge company (see other answers) they will probably not mind doing it. It definitely is not impolite or unprofessional to ask.