Mare Nostrum is just a dream,the need of a strong ally to the East that doesn't actually border you is reality.Honestly,France isn't in control of the Western Mediterranean either,Spain is a fully independent entity,not a puppet state.The term Mare Nostrum is also figurative,I take it to interpret that as an attempt to lock out the enemies of France from it.I'm not sure why Russia is seen as an enemy though.To be honest though,a strong Ottoman Empire will definitely be more useful than Russia as an ally since you can launch a pincer move on France's enemies in Central Europe.
A competitor is perhaps a better term. Besides that, Russia did beat up on Sweden, which is one of France's allies. Enemy of my enemy, and of course the Ottomans' usefulness that you've pointed out. The subsequent alliance to the Ottomans makes Russia a rival in the longer term.
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