My PhD research interest is very aligned with my advisor. However, my concern is that if I want to stay in the academia, will my advisor become my competitor in the future?
You might end up as competitors someday. I've never felt like I was in competition with any of my former students or my own advisor, but some people are hyper-competitive, and others are unlucky enough to end up in awkward situations, so your mileage may vary.
Even though it could happen, you shouldn't waste time in graduate school worrying about hypothetical future competition. There are more important things to think about, and you shouldn't let these worries interfere with learning as much as you can from your advisor.
Ultimately, becoming a successful researcher means developing your own research agenda. Over time, you should drift away from your advisor as you explore your own interests. (If you don't, it's a bad sign.) In particular, as you become an established researcher your advisor will no longer play a central role in shaping your scholarly interests, and competition with your advisor will not be much more likely or worrisome than competition with other senior people in your field.
I think most of the other answers here have missed an important point: following your PhD, you should not be staying in the same narrow subfield as your advisor. If you are directly competing for grants with your advisor, then you've done something very wrong.
The point of doctoral and postdoctoral training is to teach you to be an independent researcher. If you are doing only what your doctoral advisor did, then why do you have your own lab? You should be sufficiently distinct in your research profile that it's clear why "you are your own boss." (And if you can't come up with enough ideas to justify your own group, then you probably aren't ready to be an independent faculty member yet!)
There are really only two places academics "compete": funding and jobs. Since you are only a PhD student now, it is unlikely you will be applying for the same jobs as your advisor in the future since your advisor has such a big head start. By the time you close the gap, you will likely have made a name for yourself. As for funding, this is somewhat field dependent. For example in the US in an NIH funded field you might get a 3 year post doctoral NRSA and a 4 year k99/R00 after your PhD before you would likely be in direct competition with your advisor, and then you would have the "new investigator" benefit. Following this road you would be 12 years out before you are really in direct competition on "equal" footing. In other fields you might be competing for funding from the same pool of money earlier.
The benefit of close ties is that you can collaborate with your advisor after you finish the PhD.
It is a very nasty situation for the whole science system, when PhD student is already a potential competitor for the supervisor. This often leads to PhD studies lasting forever.
However scientific systems in most countries have more than enough measures to exclude such a competition. Most important, you frequently cannot progress from PhD student to professor inside the same institution, using the benefit that "you are already here, and everyone knows how good you are".
As a result, there is no reason for the supervisor to press down exactly you. Another competitor will come from the side anyway, and the supervisor will be with better chances after having good shared publications on your PhD project.
Competition does may happen if the professor assigns say some quite junior post doc to supervise a PhD student. Such post doc may then want to take over promising project for instance, be the first between authors, etc. However if this goes too far, it is usually possible to ask the actual professor to remove such a "supervisor" out of head.