There are a few ways of describing the difference in these sounds, but let's start with those that most people agree on.
The -a in tang is definitely not the same as in the other two syllables (ta and tan). The sound is normally written as [ɑ] and is further back than any of the other two sounds. I think most would agree with this.
The -a in tan is trickier. There seems to be some variation of how this is pronounced, varying from a normal [a] (i.e. the same as in ta) to [æ], which is further forward. It seems to be that the [a] pronunciation is more standard, but it's common to hear sounds approaching [æ] as well. You can find both transcriptions depending on where you look.
If you listen to the top row (-a, -an and -ang) on the Chinese Pronunciation Wiki, you can clearly hear the three different pronunciations (including something between [æ] and [a] for -an).
So, to answer your question, you can pronounce the -a in ta and tan more or less the same, but you should make sure that the -a in tang is further back.
The same is true for the other initials, of course, not just for t-. It's true that the initial (or any sound) influences the final (or any other sound), but I've never seen anyone claiming there's a significant difference between similar initials that learners need to worry about. These influences are often the result of transitioning from one sound to another and will happen naturally. You can certainly treat the -a in ta, da, ba, pa etc. the same.