Multipath transmission control protocol (MPTCP) is a promising transport layer protocol that enables a device to utilize multiple communication interfaces simultaneously, thereby achieving high throughput. A congestion control algorithm (CCA) employed in MPTCP constitutes a key part that controls the data flow through different subflows (SFs). There are two fundamental challenges associated with MPTCP CCAs. First, MPTCP flows should have an advantage over single-path flows; second, MPTCP flows should be fair, indicating that SFs sharing a common bottleneck should occupy a capacity fairly close to that required by a single-path flow. Several MPTCP CCAs have been developed; however, they have failed to satisfy these challenges in all scenarios. Recently, Google has introduced the bottleneck bandwidth and round-trip-time (BBR), a new CCA for single-path TCP, achieving high throughput with minimum delay by employing a network model. In the present paper, we propose a novel MPTCP CCA based on BBR named coupled multipath BBR (C-MPBBR) that satisfies the fundamental challenges by exploiting the concept of network modeling in BBR. C-MPBBR addresses the first challenge by closing the lowbandwidth SFs by tracking the delivery rate and bottleneck bandwidth (BtlBW). Then, it satisfies the second challenge through identifying those SFs that share a common bottleneck and dividing the BtlBW share corresponding to a SF among them. We implemented C-MPBBR in the Linux kernel, tested it on a wide range of scenarios by the Mininet emulation experiments, and the real-world Internet, and confirmed that the proposed C-MPBBR outperforms the existing MPTCP CCAs in terms of successfully satisfying the fundamental challenges by ensuring both throughput and fairness.