The first example is somewhat ambiguous and the meaning of 'I got you' depends on context: on the one hand it can mean 'in the absence of your boyfriend, I got you covered' or 'I got your back', as in 'take care of her', while on the other it can also mean 'now he's out of the way, I got you', in a possessive way.
(Update to address additional info provided by OP: when it comes to creating that double effect, the phrase would not change, and the mood would almost entirely be created by the way the character says it, facial expression, direction he's looking in, etc. In writing, adding ellipses, (I got you...) could have the desired result, but it usually requires some sort of introduction of such a character trait to allow the reader some context for the 'suspense' you are adding with the '...').
The short answer to your question is 'Yes, it is correct to use one instead of the other'.
However, 'got someone' and 'take care of someone' can both mean the same thing and something completely different, as the first paragraph already hinted at.
For example, 'taking care of someone' hints at a broader care (provide food, shelter, comfort, etc) than 'I got you' typically does.
Depending on the sort of care you will be providing, 'I got you' may not be the best possible wording, contextually speaking.
For example, you would not use it when taking care of someone who cannot take care of themselves, ie. providing some form of medical care.
To illustrate how context sensitive these phrases are: is the other person capable of doing something themselves, but it brings them pain, it would be OK to use 'don't worry, I got you', but if it were your grandma instead of your friend, a 'Here, let me do that' would perhaps be a more appropriate response.