Is it permissible to purchase in installments from an affiliated company that carries the same trade name as another company, 50% of which is owned by the bank, given that both companies monitor installments and offer installment plans to customers?

What is understood from the question is that the other company, which shares the same trade name, acts as a bank in providing financing. If it does so in a Sharia (definition)-compliant manner, such as through Murabaha (cost-plus financing) for the one who orders the purchase, with its Sharia-stipulated conditions, then there is no issue with its permissibility. However, if it provides loans with interest, as a usurious bank does, then the transaction is prohibited. The first company's ownership of half of the financed company does not affect this ruling.