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That's an interesting question.
Improved Familiar has a prerequisite of '... Ability to acquire a new familiar...', so I guess it comes down to whether the 'tumour familiar' is still a 'familiar' for that purpose. Personally I can't see any problem with it from that angle.
The other issue would be size - a tumour familiar is called out as being diminuative or tiny in size, whereas an earth elemental familiar is small sized. The question here is whether the tumour aspect, when on your body, is diminuative or tiny, but the familiar aspect grows to whatever size the familiar is (being all magic and stuff...), or whether no sized-changing takes place, and the tumour just detaches but otherwise remains the same (which also precludes the alchemist from taking any of the regular small-sized familiars in Ultimate Magic - unless you think having a goat growing out of your chest is cool...).
The size issue I can see as being either way. I think I'd allow the 'size changing' version, along with Improved Familiars and the like, myself - the guy's paying character resources for the discovery and the Feat, and it doesn't seem unbalanced to let him have it. I also dislike the whole 'tumour' aspect of the discovery, but would happily refluff the thing to resemble a birthmark or tattoo instead.
The idea of an imp tumor which can turn invisible and use wands is appealing and appalling all at once. Mephit tumors would be only slightly less useful as wand jockeys and with Str 13 could potentially be transformed into somewhat dangerous combatants by sharing a mutagen (and perhaps applying Beast Shape III)
I think technically per RAW, no you can`t.
I find this doubly interesting because just a while ago I realized other classes with `limited Familiar choices` (e.g. Serpent Bloodline, Bard with Parrot Familiar) in fact are just as capable of taking Improved Familiar, which then expands their Familiar Options, so I normally would have expected something similar to apply here... But it isn`t worded to allow that IMHO.
Why I don`t think so:
The ability is not called `Familiar` (that doesn`t exclude this per se, it just isn`t a supporting aspect),
and the ability says the Tumor `resembles` an animal suitable for being a Familiar,
with characteristics and abilities of that animal and those a familiar would gain,
the Tumor `acts as a Familiar`, and the abilty gives the cost to replace it.
...If it `WAS` a Familiar (as opposed to having the abilities of one), most of those wouldn`t be necessary to state, or rather, only the DIFFERENCES would be stated (i.e. Alchemist Extracts working with Share Spell, etc) rather than re-iterating standard stuff like the cost to replace. Further, the text is also specific in that it resembles ANIMALS eligible to be Familiars. Flavor-wise, the ability is written so that your Supernatural Tumour can RESEMBLE the form and abilities of Familiar Animals... That doesn`t necessarily extend to resembling Outsiders like Silvanshees, for example, by just taking a Feat.
The ability COULD have been written to be more like that of the Bard Archetype gaining a Familiar, namely explicitly proceeding from the assumption that you ARE gaining a Familiar, and then giving any restrictions on choices, or special abilities and exceptions to normal Familiar rules. Again, I was half-way expecting it work just like those other Classes with Familiar access, in which case I thought that you could take an Improved Familiar but it would be replacing the Tumor aspect`s special abilities... But it doesn`t conform to that approach, so I don`t think it works at all.
Relatedly, as far as I can tell, the Tumour Familiar does not seem to have a creature type,
along with lacking it`s own inherent Class Skills apart from Familiar list, or it`s own skill ranks.
So based on your thoughts, would it be OK for the alchemist to have a Chelestial/Fiendish/Resolute/Anarchic Animal Familiar?
Or are those not allowed too?
A tumor familiar is a familiar. So I would say yes.
As much as i like the alchemist class, i wish there had been a few less shudder causing abilities in UM. It all seemed aimed at creepy (but amazingly cool) alchemists rather than mad alchemists.