I'm not a microbiologist, but this is not how vaccines work. The vaccines are going to put "something strange" in your body that "resembles" the virus, but it's not the virus (it's either a deactivated version of it, or some proteins which identify it...) and then is YOUR immune system the one which "attacks" this pseudo-virus. In this way, your immune system is learning how to defend to "this kind" of strange agents.
In a future if your immune system sees the virus, will defend you.
If you get the two kinds at the same time, maybe the two doses are too much for your immune system to handle it properly and the side effects may affect you (e.g. you can get fever because your immune system is under too heavy load).
In any case, getting the vaccination can (maybe) actually increase your risk of a severe Covid sickness, when you are exposed to real covid virus at the same time.
https://onlinelibrary.wiley.com/doi/10.1111/ijcp.13795
And I have no data for it and I can imagine it being hard to get reliable data about it, but autoimmune sickness is on the rise. I am not sure a quickly hacked together vaccine can rule that possibility out.
There are definitely benefits to vaccine diversity. The Oxford ChAdOx1 virus involves putting a modified chimp adenovirus into people. People don't have immunity to chimp adenovirus, so it infects them and delivers the spike protein, which elicits immunity. But that might also elicit immunity to the chimp adenovirus! If that happens, then a later does of the vaccine - say, for a new strain of SARS-CoV-2 in a year or two - might not be as effective, because immunity to the chimp adenovirus will stop the vaccine delivering the spike. In that case, having a different way to deliver a vaccine will be a life-saver.