Very simple. The same arguments which lead to an IE* protolanguage, with that interpretation of facts, will also lead to a proto-language before both Proto-IE and Proto-Uralic, or even Proto-Ural-Altaic.
For "I", Greek needs another tense. For "they" the endings are different. For "he/she", the match is considered as accidental. For "thou" we have a half-good match, but for "we" and "ye" we have excellent matches.
However, Greek is IE, inside IE language family, thought to descend from Proto-IE, PIE. Finnish is Uralic, outside IE language family, thought NOT to descend from PIE. The kind of motives leading mainstream linguists (but not Trubetskoy, founder of Balkan linguistics) to accept IE group as descending from a single language are the kind of motives leading logically to accept this and Uralic had an even older common Proto-language, explaining why Finnish and Greek have so similar verb endings.
Present day Slavonic languages, as well as Germanic ones, have a verb tense system closer to Finnish than to more typically IE ones, like Greek, Latin, Sanskrit. Old Bulgarian or Church Slavonic is known to be different, but could have picked up some categories from Greek - while living close to it on Balkan.
The other idea explaining commonalities - and which would explain Finnish ones without involving an even older Proto-language - is mutual loans while languages are neighbours in a restricted geographic area. And if the spread out from Babel was gradual, that is also what we can expect.
This would also explain instant sound changes applied to words when switching from a language to another./HGL