I like mcknz's answer very much but felt I should add a little bit myself:
Some people do ask for the post title to be changed for them, and get their wish - but that typically happens in smaller companies, who are eager to attract talent. In larger companies there is usually a lot of politics involved in who is "senior", and also the pay scale might be significantly different. In other words they may not want to change the title either because:
a) They don't want to upset other devs who've been with the company longer and feel have "earned it"
b) Don't want to set themselves up to you accusing them of not paying you as much as they should for the position (they may see it as a legal liability)
c) They already have a team leader and want to make it very clear that you are a subordinate, not in charge, no matter what your responsibilities were at your old job.
Last but not least, the title is less important than actually reaching your goal of being Development Manager. I would advise further researching the company:
Check to see if there are any old/current job postings for senior developers at that firm. Look to see what the requirements are compared to the job you are interviewing for, and how your current experience compares. In other words, do you feel you would have a good shot at getting the job of Senior Dev if you had the opportunity to apply for it? They may not feel that you are ready to be Manager yet, but could you move up the ladder in a reasonable time frame, given the chance? (or do you have a lot of learning still to do?)
Check Glass Door, and other review websites. What are other former/current employees commenting about upward/lateral mobility in the company? Are they the types to encourage growth, or do they hire "automatons" whom they want to perform a certain task until the end of time? This is crucial for you to know.
This step will primarily be necessary if the first two don't turn up any decent answers. During the interview you may want to subtlety put out some feelers about the company's attitude toward employees with higher ambitions. Be careful not to give the impression that you wouldn't be happy with the position you are interviewing for. Ask questions along the lines of whether the company typically promotes employees from within their ranks vs hiring outside talent, and whether they support employees in gaining new certificates and furthering their education. Google things like "how to ask about advancement opportunities in interviews" and read up on how to best approach this conversation.
What it really comes down to is this:
They like you. They don't think you're quite manager material yet, however they want to make you part of their team, and, if you prove yourself to them, groom you to rise within their ranks.
Not So Good Scenario
They pay well, but expect people to be highly specialized at their tasks, to the point that they do not like promoting people out of a role "they excel at" (a flawed philosophy, but one which many companies still adopt, sadly). They really do like you, but not as management material - they will not offer you much in the way of upward mobility.
The only way to determine which type of company they are is to take the above steps and inform yourself. If you feel that they are a Not So Good kind of company, then maybe insist on only taking the job if the position title changes to "Senior Dev" (that way if you later jump ship it won't look like you took a step back from higher responsibility). If, however, they seem like a place with great potential, simply take the job, even if they don't give you as big of a raise as you'd like. Get your foot in the door, as they say, and climb that corporate ladder.