Nagato's got both. Madara said he's from Senju linage so being an Uzumaki counts and the eyes are Uchiha so he has both senju and Uchiha power which is required for controlling it, [
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].
Nagato doesn't have Uchiha DNA because those eyes don't possess the blood [
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]. Nagato didn't meet all the requirements (possessing the Rinnegan
and having both Uchiha and Senju DNA), so his level of control was always going to be limited [
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]. Madara said that he could only summon the statue after awakening the Rinnegan [
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]; just having the blood did nothing. Obito never showed the ability to summon the statue prior to taking the Rinnegan - only Nagato did. Nagato is the only other person (Madara aside) who could summon the Mazou [
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] - because he was the only other person who had the Rinnegan. Gedou Mazou means 'Demonic Statue of the Outer Path', it's a summon exclusive to the Rinnegan's 7th path.
So, merely possessing the Rinnegan and one bloodline allows you to summon it but gives you limited control (Nagato). Having the required bloodlines but no Rinnegan gives you proper control but you cannot summon it (Pre-Rinnegan Obito). And possessing the Rinnegan with both bloodlines will give you mastery over the statue (War-arc Obito and Madara).