You have it exactly right. Photons are fungible. Splitting a photon is just like splitting a bitcoin. It makes no sense to say whether the output coins are "the same" as the input coins.

If you produce 2 entangled photons, if you "split" it into 3, do the the 3 new photons still entangled to the first one?

I presume you meant to ask:

If you produce 2 entangled photons, and then split one of them again, do you end up with three mutually-entangled photons?

The answer to that is: yes.

The next obvious question then is: what then is the big deal about producing three mutually entangled photons? And the answer to that is that spontaneous parametric down conversion is very sensitive to the wavelength/frequency of the input light. You can't just throw in any old photon and split it. It has to be a very specific wavelength for the process to work, so as a practical matter it is nearly impossible to do the two-step process that you've asked about.